Part 6: Cross Validation

Compute Predicted Price for Model from parts 1/2

firstmodtestdata = lm(Price~LotArea+YearBuilt+BasementSF+FullBath+TotalRooms+GarageCars+WoodDeckSF, data=AmesTest6)
summary(firstmodtestdata)

Call:
lm(formula = Price ~ LotArea + YearBuilt + BasementSF + FullBath + 
    TotalRooms + GarageCars + WoodDeckSF, data = AmesTest6)

Residuals:
   Min     1Q Median     3Q    Max 
-89.40 -22.42  -3.33  18.21 205.84 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.094e+03  2.534e+02  -4.318 2.52e-05 ***
LotArea     -5.590e-06  2.603e-04  -0.021    0.983    
YearBuilt    5.382e-01  1.313e-01   4.100 6.10e-05 ***
BasementSF   5.516e-02  8.507e-03   6.484 7.36e-10 ***
FullBath     7.526e-01  7.662e+00   0.098    0.922    
TotalRooms   1.575e+01  2.512e+00   6.271 2.32e-09 ***
GarageCars   3.032e+01  4.683e+00   6.475 7.73e-10 ***
WoodDeckSF   2.667e-02  2.237e-02   1.192    0.235    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 39.9 on 192 degrees of freedom
Multiple R-squared:  0.7171,    Adjusted R-squared:  0.7068 
F-statistic: 69.52 on 7 and 192 DF,  p-value: < 2.2e-16
predict(firstmodtestdata, data.frame(AmesTest6), level = .95, interval = "predict")
          fit         lwr      upr
1   119.79432  40.5152173 199.0734
2   185.20921 103.0373915 267.3810
3   275.60888 195.7206281 355.4971
4    90.56750  10.3889986 170.7460
5    86.96406   4.8518632 169.0763
6   126.08187  46.7575130 205.4062
7   200.50411 120.4898033 280.5184
8   237.01907 157.4596418 316.5785
9   114.82438  35.5197614 194.1290
10  144.15482  64.4808637 223.8288
11  202.37219 120.6466901 284.0977
12  198.95260 119.5073488 278.3979
13  116.22655  34.9234336 197.5297
14  138.02012  57.7955372 218.2447
15  181.44156 101.4030738 261.4800
16  282.13434 202.4496567 361.8190
17  151.45066  71.9408564 230.9605
18  176.13772  96.6583594 255.6171
19  109.66909  30.2957665 189.0424
20  128.11028  48.7517242 207.4688
21  168.01417  87.9831435 248.0452
22  188.75791 108.8267372 268.6891
23  219.59739 140.2343007 298.9605
24  108.84314  29.1979588 188.4883
25  195.09081 115.5202960 274.6613
26   84.07868   4.5933893 163.5640
27  236.05029 156.6282877 315.4723
28  176.68544  97.2296266 256.1413
29  140.10917  60.5007887 219.7175
30  208.59573 129.2094646 287.9820
31  177.71860  93.4542332 261.9830
32   80.94277  -0.2203661 162.1059
33  248.78819 168.9840392 328.5923
34  177.56908  98.1220212 257.0161
35  229.49614 149.6695764 309.3227
36  235.50800 155.9280118 315.0880
37  203.26072 122.7924460 283.7290
38  268.34494 188.4888109 348.2011
39  138.62004  58.9141084 218.3260
40  185.65981 104.8518484 266.4678
41  168.23074  88.1750269 248.2865
42   55.81711 -24.3816553 136.0159
43  137.77802  57.7983272 217.7577
44  218.75158 135.8379209 301.6652
45  180.67127 101.1920290 260.1505
46  210.65405 130.4145386 290.8936
47  123.85706  43.6238691 204.0902
48  163.09484  83.4100980 242.7796
49  275.87649 195.9837153 355.7693
50  346.16506 264.4975692 427.8325
51  186.81834 107.2550225 266.3817
52  137.78640  57.0075978 218.5652
53  163.21033  83.4026336 243.0180
54  128.77159  49.3485348 208.1946
55  188.38632 106.8170245 269.9556
56  195.76089 115.9210747 275.6007
57  211.43980 131.9600581 290.9195
58  210.50282 130.8053147 290.2003
59  287.28702 207.4572322 367.1168
60  144.76171  64.8109114 224.7125
61  133.96963  54.4388225 213.5004
62  134.51164  54.9399192 214.0834
63  170.38406  90.5431180 250.2250
64  204.07838 124.4811273 283.6756
65  305.27627 225.4216817 385.1309
66  127.66096  48.0601917 207.2617
67  143.85761  64.4458487 223.2694
68  102.21825  22.4604426 181.9761
69  105.71369  26.3566438 185.0707
70  274.08487 192.8703984 355.2994
71  308.75013 227.8042068 389.6961
72  126.62414  47.0396355 206.2087
73  208.16756 128.1465377 288.1886
74  235.77503 156.3601701 315.1899
75  244.35647 164.9858838 323.7270
76  167.38927  87.9390722 246.8395
77  294.23159 213.9051280 374.5581
78  213.69345 133.8298219 293.5571
79  172.65664  92.9945005 252.3188
80  275.71954 192.7898831 358.6492
81  198.45265 115.2970060 281.6083
82   67.62563 -12.2175045 147.4688
83  129.18691  49.7081016 208.6657
84  226.65695 147.1919454 306.1220
85  201.42185 121.7322553 281.1114
86  210.20057 129.6588632 290.7423
87  245.28714 163.4744257 327.0999
88   95.52922  13.8945832 177.1639
89  141.51753  60.5590121 222.4760
90  178.89335  98.1294762 259.6572
91  145.49248  65.5823301 225.4026
92  162.67735  83.2308200 242.1239
93  104.77006  25.5319681 184.0081
94   78.00059  -1.5257067 157.5269
95  129.13558  49.6389468 208.6322
96  240.26617 160.9250730 319.6073
97  229.09864 149.2967966 308.9005
98  221.82081 142.2596961 301.3819
99  148.68380  69.1709076 228.1967
100 159.37320  79.8857695 238.8606
101 161.69776  80.6986976 242.6968
102 341.51848 260.2988486 422.7381
103 272.02799 191.2361126 352.8199
104 104.05197  24.3009462 183.8030
105  47.10138 -33.2845201 127.4873
106 216.67017 137.2407068 296.0996
107 263.24188 183.2905598 343.1932
108 119.06606  39.8509238 198.2812
109 291.59252 211.3661846 371.8189
110 105.04425  25.4875235 184.6010
111 158.37215  79.0431348 237.7012
112  76.18013  -3.7819155 156.1422
113 161.49288  80.4956209 242.4901
114  32.40084 -48.5150062 113.3167
115  94.29581  14.8171069 173.7745
116 197.12729 116.8510574 277.4035
117 165.73389  86.2411214 245.2267
118 100.31488  20.8076103 179.8221
119 107.84266  27.6557645 188.0296
120 247.50975 166.0795033 328.9400
121  36.82383 -44.2753782 117.9230
122 121.88925  41.9830717 201.7954
123 157.26566  77.7658404 236.7655
124 156.46568  75.9871461 236.9442
125 297.35013 216.0184503 378.6818
126 189.70275 109.5511470 269.8543
127 133.06156  52.9207362 213.2024
128 152.17738  72.7469421 231.6078
129  56.70398 -23.2363804 136.6443
130 129.83372  50.2603842 209.4071
131 129.36215  50.0047702 208.7195
132 145.45627  62.6684507 228.2441
133 192.60650 112.1461309 273.0669
134  72.67559  -7.6281967 152.9794
135 203.42963 124.0768193 282.7824
136 152.56893  73.1675271 231.9703
137 187.51218 107.3599896 267.6644
138 218.48547 138.3508767 298.6201
139 142.50647  61.4328906 223.5800
140 120.00407  38.3054306 201.7027
141  36.66320 -43.6243616 116.9508
142 292.34015 212.2969048 372.3834
143 203.93579 124.3287521 283.5428
144 202.21936 122.3564080 282.0823
145 140.19638  60.7979408 219.5948
146 173.89491  92.0095440 255.7803
147 103.12039  22.6092085 183.6316
148 144.99724  65.1837300 224.8108
149 213.96240 134.5627470 293.3621
150 193.89500 113.8068680 273.9831
151 142.86456  62.3486648 223.3805
152 123.00017  43.6163834 202.3840
153 240.30665 160.6690628 319.9442
154 107.84864  28.5611069 187.1362
155 166.90726  87.1000131 246.7145
156 235.32704 126.9036743 343.7504
157 215.00546 135.3627276 294.6482
158 200.81353 121.2973076 280.3298
159 136.85179  57.1292437 216.5743
160 103.14306  22.1624180 184.1237
161 274.79477 194.8895644 354.7000
162 201.41444 121.9618817 280.8670
163 231.26161 151.6439835 310.8792
164 229.01444 148.6243064 309.4046
165 180.00592  98.3362443 261.6756
166 106.26796  26.8057601 185.7302
167 177.63164  97.8459479 257.4173
168 193.92888 114.3845005 273.4733
169 210.61053 130.7533819 290.4677
170 206.93953 127.3798027 286.4993
171 133.26973  53.3533513 213.1861
172 201.90450 122.2733228 281.5357
173 150.15794  70.6435634 229.6723
174 137.96084  58.6270643 217.2946
175 111.82222  30.5850327 193.0594
176 176.50802  96.9442410 256.0718
177 277.33212 197.1893522 357.4749
178 115.51149  35.5933913 195.4296
179 144.23473  64.8254012 223.6440
180 267.63775 187.9786791 347.2968
181 151.01391  71.3849998 230.6428
182 131.01004  49.8363080 212.1838
183 101.86943  21.7993302 181.9395
184 112.88288  33.6139060 192.1518
185 210.44441 130.7608902 290.1279
186 219.43522 138.7134067 300.1570
187 147.69957  66.4253418 228.9738
188 349.31491 267.8233528 430.8065
189 152.70685  73.2617209 232.1520
190 153.43281  73.9968210 232.8688
191 203.30477 124.0193554 282.5902
192 262.86770 183.0208294 342.7146
193 140.04478  60.3634818 219.7261
194 249.00854 169.5194115 328.4977
195 174.00818  93.7265263 254.2898
196 202.52187 122.0433427 283.0004
197 201.60491 121.9709350 281.2389
198 104.06673  24.8486963 183.2848
199 219.39930 135.8095219 302.9891
200 207.69075 127.4322555 287.9492

Compute the residuals for the 200 holdout cases

rstandard(firstmod)
           1            2            3            4            5            6            7            8 
 0.873588341 -0.924906821  0.447640499 -0.027270222  1.063910057  0.174744457 -0.191277821  0.944573602 
           9           10           11           12           13           14           15           16 
 1.126501899 -1.298045855 -1.784521427 -0.390931876  0.928112004 -0.251019846 -0.967433558 -0.853587045 
          17           18           19           20           21           22           23           24 
-0.087371358 -0.409712089 -0.547687199  0.028800886 -0.755026464  0.006164081  0.894683010  0.282996269 
          25           26           27           28           29           30           31           32 
-0.838532129  0.656121242  0.479258899  0.020610155 -1.232385502 -0.217293964 -0.819525973  0.752386583 
          33           34           35           36           37           38           39           40 
 1.301730555 -0.368588576  0.012811135 -1.571498698 -1.006890504  0.321895392  0.339657829  1.348589750 
          41           42           43           44           45           46           47           48 
 0.963262369 -0.238079154  0.475724235  2.200365674 -0.105575016  1.926363924  0.042001460  0.073728503 
          49           50           51           52           53           54           55           56 
 0.285366801  5.367860526  0.004602895 -0.535364029 -0.348513231 -0.006868886 -1.351224074  0.730852565 
          57           58           59           60           61           62           63           64 
-1.048849353 -2.043381957 -0.312424719 -0.531309368 -0.334815399 -0.114328389 -0.276792206 -0.230129988 
          65           66           67           68           69           70           71           72 
 0.679733427  0.109996480 -0.454107013 -0.058890478 -0.018034749  0.723333552  1.941805102 -0.421338773 
          73           74           75           76           77           78           79           80 
-1.253381349 -0.272486984  0.774488798 -0.566457751  2.784140822 -0.602728826  0.211678634 -1.994937108 
          81           82           83           84           85           86           87           88 
-2.024610829  0.543586839 -0.105970303 -1.233809829 -0.462472098  2.511469835 -0.085901980  0.325066427 
          89           90           91           92           93           94           95           96 
-0.555573111 -0.177505261  0.494006202 -0.573718685  0.384261205  0.272235494 -0.016090265 -0.309897886 
          97           98           99          100          101          102          103          104 
-0.396482995 -1.236982818 -0.169241707 -0.085384939 -0.043860188  2.861695728 -1.494809388 -0.102922941 
         105          106          107          108          109          110          111          112 
 0.817165306 -1.180707634 -0.795678800  0.338846184  1.157144516 -0.077128355 -1.158751459  0.963346888 
         113          114          115          116          117          118          119          120 
 1.059355672  1.483725472  0.777118380 -0.566006929 -1.410876007  0.422455873  0.233959133  0.480574898 
         121          122          123          124          125          126          127          128 
 1.453286050  0.460973322 -0.057359311  1.116648142 -0.125883204 -0.761063412 -0.346267144 -0.702597768 
         129          130          131          132          133          134          135          136 
 0.707814946  1.144570875  0.091927590 -0.144639660 -1.349014738  1.338935281 -1.021587372 -0.697060989 
         137          138          139          140          141          142          143          144 
 0.123542811 -0.880443400  0.180859063  0.208614507  0.699371211  0.514068479 -0.581479506 -0.895924381 
         145          146          147          148          149          150          151          152 
-0.004965056 -1.436055172  0.073894261  0.114467801  0.089443594  0.717097685 -0.163333065 -0.202232289 
         153          154          155          156          157          158          159          160 
-0.667207525  0.130055513  0.587008268 -0.499410733  0.161432770 -0.833445933  0.079936551  0.022132570 
         161          162          163          164          165          166          167          168 
 0.410965109 -0.668316920  1.032963767 -0.999512930 -1.043325588  1.081307876  0.111010308 -0.102058043 
         169          170          171          172          173          174          175          176 
-0.524258481  3.498002418  0.725007470 -0.783755051 -0.155929312 -0.087427781 -0.474914350 -0.240915045 
         177          178          179          180          181          182          183          184 
-1.259629358 -0.906564109  0.014294054 -1.208817271 -0.570948165 -0.440510642  0.232909695  1.321785621 
         185          186          187          188          189          190          191          192 
-0.239680635  3.594317257 -0.225611359  0.798219974 -0.650350729 -0.972185513 -1.093278608 -0.454428554 
         193          194          195          196          197          198          199          200 
-0.511220334  0.341513434  0.856769820 -1.603675529  0.086104606  0.442260731 -2.398959566 -0.375693285 
rstudent(firstmod)
           1            2            3            4            5            6            7            8 
 0.873047211 -0.924557038  0.446706411 -0.027199166  1.064277615  0.174302661 -0.190797232  0.944307205 
           9           10           11           12           13           14           15           16 
 1.127296027 -1.300379520 -1.794814826 -0.390067769  0.927775423 -0.250406388 -0.967271333 -0.852981266 
          17           18           19           20           21           22           23           24 
-0.087145264 -0.408822492 -0.546686279  0.028725848 -0.754178125  0.006148009  0.894216028  0.282317223 
          25           26           27           28           29           30           31           32 
-0.837881239  0.655145245  0.478295376  0.020556435 -1.234062556 -0.216754010 -0.818822394  0.751533399 
          33           34           35           36           37           38           39           40 
 1.304103665 -0.367757591  0.012777735 -1.577579590 -1.006926956  0.321142696  0.338873974  1.351489344 
          41           42           43           44           45           46           47           48 
 0.963080547 -0.237493405  0.474763641  2.222833296 -0.105302779  1.940181703  0.041892131  0.073537292 
          49           50           51           52           53           54           55           56 
 0.284683067  5.807328060  0.004590893 -0.534367028 -0.347714461 -0.006850976 -1.354154653  0.729962907 
          57           58           59           60           61           62           63           64 
-1.049124262 -2.060582528 -0.311689289 -0.530313937 -0.334039876 -0.114034152 -0.276125549 -0.229561571 
          65           66           67           68           69           70           71           72 
 0.678778190  0.109713114 -0.453166320 -0.058737448 -0.017987737  0.722432419  1.956043914 -0.420434520 
          73           74           75           76           77           78           79           80 
-1.255258954 -0.271829022  0.773678749 -0.565453372  2.834691691 -0.601726702  0.211151308 -2.010683006 
          81           82           83           84           85           86           87           88 
-2.041238495  0.542587080 -0.105697069 -1.235500214 -0.461523302  2.547108503 -0.085679632  0.324308049 
          89           90           91           92           93           94           95           96 
-0.554570364 -0.177056933  0.493031482 -0.572713799  0.383406675  0.271578041 -0.016048319 -0.309167137 
          97           98           99          100          101          102          103          104 
-0.395611124 -1.238703042 -0.168812991 -0.085163909 -0.043746039  2.917123039 -1.499663493 -0.102657395 
         105          106          107          108          109          110          111          112 
 0.816455510 -1.181927538 -0.794915682  0.338063717  1.158172724 -0.076928429 -1.159792414  0.963165460 
         113          114          115          116          117          118          119          120 
 1.059694814  1.488414119  0.776313855 -0.565002596 -1.414548867  0.421550258  0.233382339  0.479610310 
         121          122          123          124          125          126          127          128 
 1.457535276  0.460025941 -0.057210233  1.117370580 -0.125560137 -0.760226465 -0.345472118 -0.701668294 
         129          130          131          132          133          134          135          136 
 0.706892151  1.145500975  0.091689901 -0.144270363 -1.351919320  1.341722593 -1.021704101 -0.696124755 
         137          138          139          140          141          142          143          144 
 0.123225563 -0.879925687  0.180402829  0.208094116  0.698437757  0.513081232 -0.580474602 -0.895461956 
         145          146          147          148          149          150          151          152 
-0.004952110 -1.440065247  0.073702624  0.114173214  0.089212223  0.716187523 -0.162918482 -0.201726441 
         153          154          155          156          157          158          159          160 
-0.666240549  0.129722099  0.586003686 -0.498432327  0.161022751 -0.832780481  0.079729438  0.022074886 
         161          162          163          164          165          166          167          168 
 0.410073891 -0.667350917  1.033145027 -0.999510381 -1.043567462  1.081787216  0.110724395 -0.101794681 
         169          170          171          172          173          174          175          176 
-0.523266103  3.605665651  0.724108836 -0.782964844 -0.155532564 -0.087201543 -0.473954439 -0.240323170 
         177          178          179          180          181          182          183          184 
-1.261568334 -0.906141639  0.014256789 -1.210279485 -0.569943417 -0.439584176  0.232335191  1.324378384 
         185          186          187          188          189          190          191          192 
-0.239091423  3.712004359 -0.225052896  0.797462859 -0.649370539 -0.972045937 -1.093837863 -0.453487541 
         193          194          195          196          197          198          199          200 
-0.510234672  0.340726419  0.856173966 -1.610315010  0.085881741  0.441332362 -2.429390387 -0.374851448 

Compute the mean and standard deviation of these residuals. Are they close to what you expect from the training model?

plot(firstmodtestdata$residuals~firstmodtestdata$fitted.values)
abline(a=0, b=0)

mean(firstmodtestdata$residuals)
[1] -2.966551e-16
sd(firstmodtestdata$residuals)
[1] 39.19671
hist(firstmodtestdata$residuals)

firstmodtraindata = lm(Price~LotArea+YearBuilt+BasementSF+FullBath+TotalRooms+GarageCars+WoodDeckSF, data=AmesTrain6a)
mean(firstmodtraindata$residuals)
[1] -1.342822e-16
sd(firstmodtraindata$residuals)
[1] 39.27828

The mean residuals of our simplest model using the test data are very similar to the mean residuals we had with the training data. This is a good sign that our model was not overfitted. Additionally, the standard deviations were similar, which was expected.

Are any holdout cases especially poorly predicted by the training model? If so, identify by the row number(s) in the holdout data.

which.max(rstandard(firstmodtestdata))
50 
50 
max(rstandard(firstmodtestdata))
[1] 5.367861
which.max(rstudent(firstmodtestdata))
50 
50 
max(rstudent(firstmodtestdata))
[1] 5.807328
which.min(rstandard(firstmodtestdata))
199 
199 
min(rstandard(firstmodtestdata))
[1] -2.39896
which.min(rstudent(firstmodtestdata))
199 
199 
min(rstudent(firstmodtestdata))
[1] -2.42939

Although some of these outliers are over the threshold for concern for rstudent and rstandard residuals, they are not substantially different from the outliers we had in our first model. This means that overfitting is likely not the cause of the outliers of concern. Finally, the outliers don’t change substantially between rstandard and rstudent, meaning that their removal from the data set doesn’t significantly alter our model.

Compute the correlation between the predicted values above and actual prices for the holdout sample.

summary(firstmodtraindata)

Call:
lm(formula = Price ~ LotArea + YearBuilt + BasementSF + FullBath + 
    TotalRooms + GarageCars + WoodDeckSF, data = AmesTrain6a)

Residuals:
     Min       1Q   Median       3Q      Max 
-143.676  -22.289   -3.192   16.325  223.801 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.418e+03  1.390e+02 -10.197  < 2e-16 ***
LotArea      1.753e-03  4.293e-04   4.084 5.04e-05 ***
YearBuilt    7.127e-01  7.154e-02   9.962  < 2e-16 ***
BasementSF   4.947e-02  4.610e-03  10.731  < 2e-16 ***
FullBath     8.294e+00  4.097e+00   2.025   0.0434 *  
TotalRooms   1.091e+01  1.412e+00   7.723 4.92e-14 ***
GarageCars   1.810e+01  2.885e+00   6.274 6.85e-10 ***
WoodDeckSF   6.867e-02  1.453e-02   4.725 2.88e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 39.51 on 587 degrees of freedom
Multiple R-squared:  0.6833,    Adjusted R-squared:  0.6795 
F-statistic: 180.9 on 7 and 587 DF,  p-value: < 2.2e-16
fitAmes=predict(firstmodtraindata, newdata=AmesTest6)
holoutresid=(AmesTest6$Price)-fitAmes
mean(holdoutresid)
[1] 0.01494272
cor(AmesTest6$Price, fitAmes)
[1] 0.7888105
crosscorr=cor(AmesTest6$Price, fitAmes)
crosscorr^2
[1] 0.622222
0.6795-crosscorr^2
[1] 0.05727804

Our shrinkage is 0.05727804, which indicates that our model fits our test data almost as well as it fitr our training data. This means that we did not overfit our model (which makes sense, because this was the most basic model that we used).

Part 7: A Fancy Model

Categorical variables from the original dataset

modTransformCat=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCat)

Call:
lm(formula = Price ~ factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(HeatingQC) + factor(KitchenQ) + 
    factor(ExteriorC) + factor(CentralAir) + factor(GarageQ) + 
    factor(Foundation) + factor(GarageC) + factor(BasementHt) + 
    factor(GarageType) + factor(LotConfig) + factor(BasementC) + 
    factor(Heating) + factor(Condition), data = AmesTrain6a)

Residuals:
    Min      1Q  Median      3Q     Max 
-96.316 -23.460  -0.895  18.138 154.613 

Coefficients: (4 not defined because of singularities)
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)                276.8311    87.8232   3.152 0.001712 ** 
factor(HouseStyle)1.5Unf    -4.5273    20.4211  -0.222 0.824633    
factor(HouseStyle)1Story    -0.0779     6.5346  -0.012 0.990493    
factor(HouseStyle)2.5Unf    42.7315    23.8827   1.789 0.074150 .  
factor(HouseStyle)2Story    11.6310     7.0351   1.653 0.098862 .  
factor(HouseStyle)SFoyer   -24.9542    12.1385  -2.056 0.040291 *  
factor(HouseStyle)SLvl      -7.8712     9.2634  -0.850 0.395873    
factor(ExteriorQ)Fa       -127.2075    23.2606  -5.469 6.99e-08 ***
factor(ExteriorQ)Gd        -62.8435    13.3102  -4.721 3.00e-06 ***
factor(ExteriorQ)TA        -94.3985    14.1932  -6.651 7.27e-11 ***
factor(BasementFin)BLQ       0.8396     7.0710   0.119 0.905529    
factor(BasementFin)GLQ       3.5429     5.5164   0.642 0.520991    
factor(BasementFin)LwQ       1.7849     8.8851   0.201 0.840869    
factor(BasementFin)None   -110.4109    42.1347  -2.620 0.009034 ** 
factor(BasementFin)Rec       1.2895     7.3736   0.175 0.861240    
factor(BasementFin)Unf      -6.0051     5.3489  -1.123 0.262085    
factor(HeatingQC)Fa         -0.6798    11.2329  -0.061 0.951765    
factor(HeatingQC)Gd          4.0802     5.3941   0.756 0.449737    
factor(HeatingQC)TA         -1.1773     4.9969  -0.236 0.813823    
factor(KitchenQ)Fa         -52.7203    15.1922  -3.470 0.000562 ***
factor(KitchenQ)Gd         -34.8167     9.5749  -3.636 0.000304 ***
factor(KitchenQ)TA         -40.5129    10.1530  -3.990 7.53e-05 ***
factor(ExteriorC)Fa          4.3272    31.9069   0.136 0.892173    
factor(ExteriorC)Gd        -25.8265    26.4996  -0.975 0.330203    
factor(ExteriorC)TA        -19.6775    27.3093  -0.721 0.471509    
factor(CentralAir)Y          5.2874     8.6428   0.612 0.540953    
factor(GarageQ)Gd           61.5677    27.1955   2.264 0.023984 *  
factor(GarageQ)None         46.7622    50.8032   0.920 0.357753    
factor(GarageQ)Po           49.5994    70.6183   0.702 0.482764    
factor(GarageQ)TA            0.1246    10.8491   0.011 0.990844    
factor(Foundation)CBlock     1.7521     7.6006   0.231 0.817772    
factor(Foundation)PConc     11.4451     8.6758   1.319 0.187676    
factor(Foundation)Slab      12.2294    22.4476   0.545 0.586120    
factor(Foundation)Stone     18.3872    24.4244   0.753 0.451891    
factor(Foundation)Wood      24.3737    29.0463   0.839 0.401774    
factor(GarageC)Fa           85.3064    48.2085   1.770 0.077380 .  
factor(GarageC)Gd           98.6830    50.8363   1.941 0.052765 .  
factor(GarageC)None              NA         NA      NA       NA    
factor(GarageC)Po           88.2301    55.9473   1.577 0.115387    
factor(GarageC)TA           98.0079    46.5230   2.107 0.035617 *  
factor(BasementHt)Fa       -89.7863    14.3966  -6.237 9.14e-10 ***
factor(BasementHt)Gd       -58.4549     8.2455  -7.089 4.33e-12 ***
factor(BasementHt)None           NA         NA      NA       NA    
factor(BasementHt)TA       -72.2207    10.0079  -7.216 1.86e-12 ***
factor(GarageType)Attchd   -17.1134    18.0483  -0.948 0.343460    
factor(GarageType)Basment  -32.2733    23.4157  -1.378 0.168700    
factor(GarageType)BuiltIn   -5.6891    19.0984  -0.298 0.765907    
factor(GarageType)CarPort  -58.3085    45.1565  -1.291 0.197178    
factor(GarageType)Detchd   -46.0371    18.1132  -2.542 0.011317 *  
factor(GarageType)None           NA         NA      NA       NA    
factor(LotConfig)CulDSac     7.4004     7.6958   0.962 0.336682    
factor(LotConfig)FR2       -20.1713     8.8796  -2.272 0.023509 *  
factor(LotConfig)FR3        10.9377    20.5572   0.532 0.594905    
factor(LotConfig)Inside     -7.3204     4.6136  -1.587 0.113172    
factor(BasementC)Fa         -7.9727    39.4513  -0.202 0.839924    
factor(BasementC)Gd         -1.8171    39.2841  -0.046 0.963124    
factor(BasementC)None            NA         NA      NA       NA    
factor(BasementC)TA        -18.5301    38.3507  -0.483 0.629172    
factor(Heating)GasW         24.5236    17.0449   1.439 0.150806    
factor(Heating)Grav        -43.9947    45.3212  -0.971 0.332125    
factor(Heating)OthW         -1.8212    41.8501  -0.044 0.965306    
factor(Heating)Wall          8.0581    32.3359   0.249 0.803302    
factor(Condition)3           0.4042    51.1716   0.008 0.993701    
factor(Condition)4          22.0054    50.2147   0.438 0.661400    
factor(Condition)5          35.7435    50.0151   0.715 0.475137    
factor(Condition)6          33.5797    50.1292   0.670 0.503237    
factor(Condition)7          36.8567    50.2968   0.733 0.464014    
factor(Condition)8          44.5512    50.5200   0.882 0.378256    
factor(Condition)9          49.7814    52.3594   0.951 0.342158    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 37.57 on 530 degrees of freedom
Multiple R-squared:  0.7414,    Adjusted R-squared:  0.7102 
F-statistic: 23.75 on 64 and 530 DF,  p-value: < 2.2e-16
MSE=(summary(modTransformCat)$sigma)^2
step(none,scope=list(upper=modTransformCat),scale=MSE)
Start:  AIC=1456.74
Price ~ 1

                      Df Sum of Sq     RSS      Cp
+ factor(ExteriorQ)    3   1606904 1286545  324.40
+ factor(BasementHt)   4   1555913 1337536  362.52
+ factor(KitchenQ)     3   1305178 1588271  538.14
+ factor(Foundation)   5    973531 1919917  777.09
+ factor(GarageType)   6    737589 2155859  946.23
+ factor(HeatingQC)    3    635156 2258293 1012.79
+ factor(Condition)    7    546078 2347371 1083.90
+ factor(BasementFin)  6    515220 2378228 1103.76
+ factor(HouseStyle)   6    230783 2662666 1305.26
+ factor(GarageC)      5    220424 2673025 1310.59
+ factor(GarageQ)      4    213737 2679712 1313.33
+ factor(CentralAir)   1    171099 2722350 1337.54
+ factor(BasementC)    4    116360 2777088 1382.31
+ factor(ExteriorC)    3     73758 2819691 1410.49
+ factor(LotConfig)    4     40148 2853301 1436.30
+ factor(Heating)      4     31951 2861498 1442.11
<none>                             2893449 1456.74

Step:  AIC=324.4
Price ~ factor(ExteriorQ)

                      Df Sum of Sq     RSS      Cp
+ factor(BasementHt)   4    250784 1035761  154.74
+ factor(GarageType)   6    213599 1072946  185.08
+ factor(Foundation)   5    127908 1158637  243.79
+ factor(KitchenQ)     3    111799 1174746  251.20
+ factor(GarageC)      5     61878 1224668  290.57
+ factor(CentralAir)   1     49963 1236582  291.01
+ factor(GarageQ)      4     51506 1235039  295.91
+ factor(Condition)    7     56831 1229715  298.14
+ factor(HouseStyle)   6     51999 1234546  299.56
+ factor(BasementFin)  6     50313 1236232  300.76
+ factor(HeatingQC)    3     34665 1251880  305.84
+ factor(LotConfig)    4     37324 1249222  305.96
+ factor(BasementC)    4     31824 1254721  309.86
+ factor(Heating)      4     14719 1271826  321.97
+ factor(ExteriorC)    3      9120 1277425  323.94
<none>                             1286545  324.40
- factor(ExteriorQ)    3   1606904 2893449 1456.74

Step:  AIC=154.74
Price ~ factor(ExteriorQ) + factor(BasementHt)

                      Df Sum of Sq     RSS      Cp
+ factor(GarageType)   6    124591  911170  78.481
+ factor(KitchenQ)     3     58853  976908 119.050
+ factor(GarageQ)      4     38749  997012 135.292
+ factor(HouseStyle)   6     43956  991805 135.604
+ factor(GarageC)      5     38839  996922 137.228
+ factor(CentralAir)   1     23287 1012474 140.246
+ factor(Foundation)   5     32902 1002859 141.434
+ factor(LotConfig)    4     24518 1011243 145.374
+ factor(HeatingQC)    3     19358 1016403 147.029
+ factor(Condition)    7     29957 1005804 147.521
<none>                             1035761 154.742
+ factor(ExteriorC)    3      6387 1029374 156.217
+ factor(Heating)      4      9177 1026584 156.241
+ factor(BasementC)    3      5603 1030159 156.773
+ factor(BasementFin)  5      9762 1025999 157.827
- factor(BasementHt)   4    250784 1286545 324.400
- factor(ExteriorQ)    3    301775 1337536 362.522

Step:  AIC=78.48
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType)

                      Df Sum of Sq     RSS      Cp
+ factor(KitchenQ)     3     46077  865093  51.839
+ factor(HouseStyle)   6     33520  877650  66.735
+ factor(Condition)    7     30314  880856  71.006
+ factor(LotConfig)    4     17289  893881  74.233
+ factor(GarageQ)      3     13232  897938  75.107
+ factor(BasementC)    3     11289  899881  76.484
+ factor(CentralAir)   1      4507  906662  77.288
+ factor(HeatingQC)    3      9175  901995  77.981
+ factor(ExteriorC)    3      8888  902282  78.184
+ factor(Foundation)   5     14356  896814  78.311
<none>                              911170  78.481
+ factor(GarageC)      4      7043  904127  81.491
+ factor(Heating)      4      5425  905744  82.637
+ factor(BasementFin)  5      6296  904874  84.021
- factor(GarageType)   6    124591 1035761 154.742
- factor(BasementHt)   4    161777 1072946 185.085
- factor(ExteriorQ)    3    262738 1173908 258.607

Step:  AIC=51.84
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ)

                      Df Sum of Sq    RSS      Cp
+ factor(HouseStyle)   6     29516 835576  42.930
+ factor(LotConfig)    4     16523 848569  48.134
+ factor(Condition)    7     20942 844151  51.004
+ factor(BasementC)    3      9422 855671  51.165
<none>                             865093  51.839
+ factor(CentralAir)   1      2384 862708  52.150
+ factor(GarageQ)      3      7667 857426  52.408
+ factor(Foundation)   5     11629 853463  53.601
+ factor(ExteriorC)    3      4888 860204  54.376
+ factor(GarageC)      4      7585 857508  54.466
+ factor(HeatingQC)    3      2954 862139  55.747
+ factor(Heating)      4      3609 861484  57.283
+ factor(BasementFin)  5      4771 860322  58.459
- factor(KitchenQ)     3     46077 911170  78.481
- factor(GarageType)   6    111815 976908 119.050
- factor(ExteriorQ)    3    111470 976562 124.805
- factor(BasementHt)   4    131934 997027 137.303

Step:  AIC=42.93
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle)

                      Df Sum of Sq    RSS      Cp
+ factor(LotConfig)    4     19231 816346  37.306
+ factor(BasementC)    3     10072 825504  41.795
+ factor(Condition)    7     21361 814215  41.797
+ factor(CentralAir)   1      3577 832000  42.396
<none>                             835576  42.930
+ factor(GarageQ)      3      7821 827756  43.389
+ factor(GarageC)      4      7300 828276  45.758
+ factor(ExteriorC)    3      3909 831667  46.160
+ factor(BasementFin)  5      8833 826743  46.672
+ factor(Foundation)   5      8749 826827  46.732
+ factor(HeatingQC)    3      2709 832867  47.010
+ factor(Heating)      4      2844 832732  48.915
- factor(HouseStyle)   6     29516 865093  51.839
- factor(KitchenQ)     3     42073 877650  66.735
- factor(GarageType)   6    103598 939174 104.319
- factor(ExteriorQ)    3    103129 938705 109.987
- factor(BasementHt)   4    135911 971487 131.210

Step:  AIC=37.31
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle) + factor(LotConfig)

                      Df Sum of Sq    RSS      Cp
+ factor(Condition)    7     21085 795261  36.370
+ factor(BasementC)    3      9549 806796  36.542
+ factor(CentralAir)   1      3152 813193  37.073
<none>                             816346  37.306
+ factor(GarageQ)      3      5790 810556  39.205
+ factor(BasementFin)  5      9685 806661  40.446
+ factor(ExteriorC)    3      3928 812418  40.524
+ factor(Foundation)   5      9138 807207  40.833
+ factor(GarageC)      4      6012 810334  41.048
+ factor(HeatingQC)    3      2690 813656  41.401
- factor(LotConfig)    4     19231 835576  42.930
+ factor(Heating)      4      2312 814033  43.669
- factor(HouseStyle)   6     32224 848569  48.134
- factor(KitchenQ)     3     41042 857388  60.381
- factor(GarageType)   6     98041 914387  94.760
- factor(ExteriorQ)    3    107192 923538 107.243
- factor(BasementHt)   4    128373 944719 120.247

Step:  AIC=36.37
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle) + factor(LotConfig) + 
    factor(Condition)

                      Df Sum of Sq    RSS      Cp
<none>                             795261  36.370
+ factor(BasementC)    3      7759 787502  36.873
- factor(Condition)    7     21085 816346  37.306
+ factor(CentralAir)   1       825 794436  37.786
+ factor(GarageQ)      3      5350 789911  38.580
+ factor(Foundation)   5     10768 784493  38.742
+ factor(ExteriorC)    3      3198 792063  40.104
+ factor(BasementFin)  5      8249 787012  40.526
+ factor(HeatingQC)    3      1677 793584  41.182
+ factor(GarageC)      4      3844 791417  41.647
- factor(LotConfig)    4     18954 814215  41.797
+ factor(Heating)      4      2858 792403  42.345
- factor(HouseStyle)   6     32660 827921  47.506
- factor(KitchenQ)     3     33199 828460  53.889
- factor(GarageType)   6     98676 893937  94.273
- factor(ExteriorQ)    3     97091 892352  99.150
- factor(BasementHt)   4    126197 921458 117.769

Call:
lm(formula = Price ~ factor(ExteriorQ) + factor(BasementHt) + 
    factor(GarageType) + factor(KitchenQ) + factor(HouseStyle) + 
    factor(LotConfig) + factor(Condition), data = AmesTrain6a)

Coefficients:
              (Intercept)        factor(ExteriorQ)Fa        factor(ExteriorQ)Gd        factor(ExteriorQ)TA  
                  371.587                   -127.471                    -64.446                    -97.138  
     factor(BasementHt)Fa       factor(BasementHt)Gd     factor(BasementHt)None       factor(BasementHt)TA  
                 -101.217                    -61.932                    -93.392                    -80.750  
 factor(GarageType)Attchd  factor(GarageType)Basment  factor(GarageType)BuiltIn  factor(GarageType)CarPort  
                  -15.095                    -33.555                     -7.098                    -67.326  
 factor(GarageType)Detchd     factor(GarageType)None         factor(KitchenQ)Fa         factor(KitchenQ)Gd  
                  -46.666                    -47.290                    -55.587                    -34.705  
       factor(KitchenQ)TA   factor(HouseStyle)1.5Unf   factor(HouseStyle)1Story   factor(HouseStyle)2.5Unf  
                  -43.160                    -12.327                     -2.912                     32.865  
 factor(HouseStyle)2Story   factor(HouseStyle)SFoyer     factor(HouseStyle)SLvl   factor(LotConfig)CulDSac  
                    9.627                    -28.005                     -8.037                      9.395  
     factor(LotConfig)FR2       factor(LotConfig)FR3    factor(LotConfig)Inside         factor(Condition)3  
                  -20.856                     12.616                     -7.700                     -6.741  
       factor(Condition)4         factor(Condition)5         factor(Condition)6         factor(Condition)7  
                    7.656                     22.628                     20.582                     24.883  
       factor(Condition)8         factor(Condition)9  
                   32.250                     41.650  
modCatTransformForward=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
MSE=(summary(modCatTransformForward)$sigma)^2
none=lm(Price~1,data=AmesTrain6a)
step(none,scope=list(upper=modCatTransformForward),scale=MSE, direction = "forward")
Start:  AIC=1456.74
Price ~ 1

                      Df Sum of Sq     RSS      Cp
+ factor(ExteriorQ)    3   1606904 1286545  324.40
+ factor(BasementHt)   4   1555913 1337536  362.52
+ factor(KitchenQ)     3   1305178 1588271  538.14
+ factor(Foundation)   5    973531 1919917  777.09
+ factor(GarageType)   6    737589 2155859  946.23
+ factor(HeatingQC)    3    635156 2258293 1012.79
+ factor(Condition)    7    546078 2347371 1083.90
+ factor(BasementFin)  6    515220 2378228 1103.76
+ factor(HouseStyle)   6    230783 2662666 1305.26
+ factor(GarageC)      5    220424 2673025 1310.59
+ factor(GarageQ)      4    213737 2679712 1313.33
+ factor(CentralAir)   1    171099 2722350 1337.54
+ factor(BasementC)    4    116360 2777088 1382.31
+ factor(ExteriorC)    3     73758 2819691 1410.49
+ factor(LotConfig)    4     40148 2853301 1436.30
+ factor(Heating)      4     31951 2861498 1442.11
<none>                             2893449 1456.74

Step:  AIC=324.4
Price ~ factor(ExteriorQ)

                      Df Sum of Sq     RSS     Cp
+ factor(BasementHt)   4    250784 1035761 154.74
+ factor(GarageType)   6    213599 1072946 185.08
+ factor(Foundation)   5    127908 1158637 243.79
+ factor(KitchenQ)     3    111799 1174746 251.20
+ factor(GarageC)      5     61878 1224668 290.57
+ factor(CentralAir)   1     49963 1236582 291.01
+ factor(GarageQ)      4     51506 1235039 295.91
+ factor(Condition)    7     56831 1229715 298.14
+ factor(HouseStyle)   6     51999 1234546 299.56
+ factor(BasementFin)  6     50313 1236232 300.76
+ factor(HeatingQC)    3     34665 1251880 305.84
+ factor(LotConfig)    4     37324 1249222 305.96
+ factor(BasementC)    4     31824 1254721 309.86
+ factor(Heating)      4     14719 1271826 321.97
+ factor(ExteriorC)    3      9120 1277425 323.94
<none>                             1286545 324.40

Step:  AIC=154.74
Price ~ factor(ExteriorQ) + factor(BasementHt)

                      Df Sum of Sq     RSS      Cp
+ factor(GarageType)   6    124591  911170  78.481
+ factor(KitchenQ)     3     58853  976908 119.050
+ factor(GarageQ)      4     38749  997012 135.292
+ factor(HouseStyle)   6     43956  991805 135.604
+ factor(GarageC)      5     38839  996922 137.228
+ factor(CentralAir)   1     23287 1012474 140.246
+ factor(Foundation)   5     32902 1002859 141.434
+ factor(LotConfig)    4     24518 1011243 145.374
+ factor(HeatingQC)    3     19358 1016403 147.029
+ factor(Condition)    7     29957 1005804 147.521
<none>                             1035761 154.742
+ factor(ExteriorC)    3      6387 1029374 156.217
+ factor(Heating)      4      9177 1026584 156.241
+ factor(BasementC)    3      5603 1030159 156.773
+ factor(BasementFin)  5      9762 1025999 157.827

Step:  AIC=78.48
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType)

                      Df Sum of Sq    RSS     Cp
+ factor(KitchenQ)     3     46077 865093 51.839
+ factor(HouseStyle)   6     33520 877650 66.735
+ factor(Condition)    7     30314 880856 71.006
+ factor(LotConfig)    4     17289 893881 74.233
+ factor(GarageQ)      3     13232 897938 75.107
+ factor(BasementC)    3     11289 899881 76.484
+ factor(CentralAir)   1      4507 906662 77.288
+ factor(HeatingQC)    3      9175 901995 77.981
+ factor(ExteriorC)    3      8888 902282 78.184
+ factor(Foundation)   5     14356 896814 78.311
<none>                             911170 78.481
+ factor(GarageC)      4      7043 904127 81.491
+ factor(Heating)      4      5425 905744 82.637
+ factor(BasementFin)  5      6296 904874 84.021

Step:  AIC=51.84
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ)

                      Df Sum of Sq    RSS     Cp
+ factor(HouseStyle)   6   29516.5 835576 42.930
+ factor(LotConfig)    4   16523.4 848569 48.134
+ factor(Condition)    7   20942.2 844151 51.004
+ factor(BasementC)    3    9421.5 855671 51.165
<none>                             865093 51.839
+ factor(CentralAir)   1    2384.5 862708 52.150
+ factor(GarageQ)      3    7666.8 857426 52.408
+ factor(Foundation)   5   11629.4 853463 53.601
+ factor(ExteriorC)    3    4888.3 860204 54.376
+ factor(GarageC)      4    7585.1 857508 54.466
+ factor(HeatingQC)    3    2954.0 862139 55.747
+ factor(Heating)      4    3609.1 861484 57.283
+ factor(BasementFin)  5    4771.0 860322 58.459

Step:  AIC=42.93
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle)

                      Df Sum of Sq    RSS     Cp
+ factor(LotConfig)    4   19230.6 816346 37.306
+ factor(BasementC)    3   10071.9 825504 41.795
+ factor(Condition)    7   21361.3 814215 41.797
+ factor(CentralAir)   1    3576.7 832000 42.396
<none>                             835576 42.930
+ factor(GarageQ)      3    7820.7 827756 43.389
+ factor(GarageC)      4    7299.8 828276 45.758
+ factor(ExteriorC)    3    3909.3 831667 46.160
+ factor(BasementFin)  5    8833.3 826743 46.672
+ factor(Foundation)   5    8749.2 826827 46.732
+ factor(HeatingQC)    3    2709.1 832867 47.010
+ factor(Heating)      4    2843.8 832732 48.915

Step:  AIC=37.31
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle) + factor(LotConfig)

                      Df Sum of Sq    RSS     Cp
+ factor(Condition)    7   21084.9 795261 36.370
+ factor(BasementC)    3    9549.4 806796 36.542
+ factor(CentralAir)   1    3152.2 813193 37.073
<none>                             816346 37.306
+ factor(GarageQ)      3    5789.7 810556 39.205
+ factor(BasementFin)  5    9684.9 806661 40.446
+ factor(ExteriorC)    3    3928.0 812418 40.524
+ factor(Foundation)   5    9138.4 807207 40.833
+ factor(GarageC)      4    6011.9 810334 41.048
+ factor(HeatingQC)    3    2689.6 813656 41.401
+ factor(Heating)      4    2312.3 814033 43.669

Step:  AIC=36.37
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle) + factor(LotConfig) + 
    factor(Condition)

                      Df Sum of Sq    RSS     Cp
<none>                             795261 36.370
+ factor(BasementC)    3    7759.2 787502 36.873
+ factor(CentralAir)   1     824.6 794436 37.786
+ factor(GarageQ)      3    5350.1 789911 38.580
+ factor(Foundation)   5   10767.8 784493 38.742
+ factor(ExteriorC)    3    3198.1 792063 40.104
+ factor(BasementFin)  5    8249.1 787012 40.526
+ factor(HeatingQC)    3    1677.2 793584 41.182
+ factor(GarageC)      4    3843.9 791417 41.647
+ factor(Heating)      4    2857.7 792403 42.345

Call:
lm(formula = Price ~ factor(ExteriorQ) + factor(BasementHt) + 
    factor(GarageType) + factor(KitchenQ) + factor(HouseStyle) + 
    factor(LotConfig) + factor(Condition), data = AmesTrain6a)

Coefficients:
              (Intercept)        factor(ExteriorQ)Fa        factor(ExteriorQ)Gd        factor(ExteriorQ)TA  
                  371.587                   -127.471                    -64.446                    -97.138  
     factor(BasementHt)Fa       factor(BasementHt)Gd     factor(BasementHt)None       factor(BasementHt)TA  
                 -101.217                    -61.932                    -93.392                    -80.750  
 factor(GarageType)Attchd  factor(GarageType)Basment  factor(GarageType)BuiltIn  factor(GarageType)CarPort  
                  -15.095                    -33.555                     -7.098                    -67.326  
 factor(GarageType)Detchd     factor(GarageType)None         factor(KitchenQ)Fa         factor(KitchenQ)Gd  
                  -46.666                    -47.290                    -55.587                    -34.705  
       factor(KitchenQ)TA   factor(HouseStyle)1.5Unf   factor(HouseStyle)1Story   factor(HouseStyle)2.5Unf  
                  -43.160                    -12.327                     -2.912                     32.865  
 factor(HouseStyle)2Story   factor(HouseStyle)SFoyer     factor(HouseStyle)SLvl   factor(LotConfig)CulDSac  
                    9.627                    -28.005                     -8.037                      9.395  
     factor(LotConfig)FR2       factor(LotConfig)FR3    factor(LotConfig)Inside         factor(Condition)3  
                  -20.856                     12.616                     -7.700                     -6.741  
       factor(Condition)4         factor(Condition)5         factor(Condition)6         factor(Condition)7  
                    7.656                     22.628                     20.582                     24.883  
       factor(Condition)8         factor(Condition)9  
                   32.250                     41.650  
modTransformCatBackward=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
MSE=(summary(modTransformCatBackward)$sigma)^2
step(modTransformCatBackward,scale=MSE)
Start:  AIC=65
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(BasementFin) + 
    factor(HeatingQC) + factor(KitchenQ) + factor(ExteriorC) + 
    factor(CentralAir) + factor(GarageQ) + factor(Foundation) + 
    factor(GarageC) + factor(BasementHt) + factor(GarageType) + 
    factor(LotConfig) + factor(BasementC) + factor(Heating) + 
    factor(Condition)

                      Df Sum of Sq    RSS      Cp
- factor(Foundation)   5      5666 753821  59.014
- factor(BasementFin)  5      6638 754793  59.702
- factor(HeatingQC)    3      1468 749623  60.040
- factor(Heating)      4      4527 752683  60.207
- factor(GarageC)      4      7684 755840  62.444
- factor(ExteriorC)    3      5741 753897  63.067
- factor(Condition)    7     17183 765339  63.173
- factor(CentralAir)   1       528 748684  63.374
- factor(BasementC)    3      7457 755613  64.283
<none>                             748156  65.000
- factor(GarageQ)      3      9357 757513  65.629
- factor(LotConfig)    4     14998 763154  67.625
- factor(HouseStyle)   6     30084 778239  74.312
- factor(KitchenQ)     3     24715 772870  76.508
- factor(GarageType)   5     66239 814395 101.925
- factor(ExteriorQ)    3     82469 830624 117.421
- factor(BasementHt)   3     84383 832539 118.778

Step:  AIC=59.01
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(BasementFin) + 
    factor(HeatingQC) + factor(KitchenQ) + factor(ExteriorC) + 
    factor(CentralAir) + factor(GarageQ) + factor(GarageC) + 
    factor(BasementHt) + factor(GarageType) + factor(LotConfig) + 
    factor(BasementC) + factor(Heating) + factor(Condition)

                      Df Sum of Sq    RSS      Cp
- factor(BasementFin)  5      6861 760682  53.874
- factor(HeatingQC)    3      1864 755686  54.334
- factor(Heating)      4      6243 760065  55.437
- factor(GarageC)      4      8523 762344  57.051
- factor(CentralAir)   1       784 754605  57.569
- factor(Condition)    7     18067 771889  57.813
- factor(ExteriorC)    3      6914 760736  57.912
- factor(BasementC)    3      8469 762290  59.013
<none>                             753821  59.014
- factor(GarageQ)      3     10841 764662  60.693
- factor(LotConfig)    4     14638 768459  61.383
- factor(HouseStyle)   6     33978 787799  71.084
- factor(KitchenQ)     3     25739 779560  71.247
- factor(GarageType)   5     71777 825598  99.861
- factor(ExteriorQ)    3     84305 838126 112.736
- factor(BasementHt)   3     96056 849877 121.060

Step:  AIC=53.87
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(HeatingQC) + 
    factor(KitchenQ) + factor(ExteriorC) + factor(CentralAir) + 
    factor(GarageQ) + factor(GarageC) + factor(BasementHt) + 
    factor(GarageType) + factor(LotConfig) + factor(BasementC) + 
    factor(Heating) + factor(Condition)

                     Df Sum of Sq    RSS      Cp
- factor(HeatingQC)   3      1799 762481  49.148
- factor(Heating)     4      6187 766869  50.257
- factor(GarageC)     4      8164 768845  51.657
- factor(CentralAir)  1      1036 761717  52.607
- factor(ExteriorC)   3      7228 767909  52.994
- factor(Condition)   7     19434 780115  53.641
<none>                            760682  53.874
- factor(BasementC)   3     10045 770726  54.989
- factor(GarageQ)     3     10341 771023  55.199
- factor(LotConfig)   4     14118 774799  55.875
- factor(HouseStyle)  6     29978 790659  63.110
- factor(KitchenQ)    3     27319 788001  67.227
- factor(GarageType)  5     73113 833795  95.668
- factor(ExteriorQ)   3     84841 845523 107.976
- factor(BasementHt)  3    107894 868575 124.306

Step:  AIC=49.15
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(KitchenQ) + 
    factor(ExteriorC) + factor(CentralAir) + factor(GarageQ) + 
    factor(GarageC) + factor(BasementHt) + factor(GarageType) + 
    factor(LotConfig) + factor(BasementC) + factor(Heating) + 
    factor(Condition)

                     Df Sum of Sq    RSS      Cp
- factor(Heating)     4      6115 768596  45.480
- factor(GarageC)     4      7797 770278  46.672
- factor(ExteriorC)   3      7070 769551  48.157
- factor(CentralAir)  1      1477 763958  48.195
- factor(Condition)   7     19403 781884  48.894
<none>                            762481  49.148
- factor(BasementC)   3      9962 772444  50.206
- factor(GarageQ)     3     10426 772907  50.534
- factor(LotConfig)   4     14161 776642  51.180
- factor(HouseStyle)  6     31376 793857  59.376
- factor(KitchenQ)    3     29453 791935  64.013
- factor(GarageType)  5     74425 836906  91.871
- factor(ExteriorQ)   3     88671 851152 105.963
- factor(BasementHt)  3    108927 871408 120.313

Step:  AIC=45.48
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(KitchenQ) + 
    factor(ExteriorC) + factor(CentralAir) + factor(GarageQ) + 
    factor(GarageC) + factor(BasementHt) + factor(GarageType) + 
    factor(LotConfig) + factor(BasementC) + factor(Condition)

                     Df Sum of Sq    RSS      Cp
- factor(GarageC)     4      7439 776035  42.750
- factor(ExteriorC)   3      4953 773549  42.989
- factor(CentralAir)  1       515 769111  43.845
- factor(Condition)   7     17492 786088  43.872
<none>                            768596  45.480
- factor(GarageQ)     3      9250 777846  46.033
- factor(BasementC)   3     10096 778692  46.632
- factor(LotConfig)   4     15422 784018  48.405
- factor(HouseStyle)  6     31180 799776  55.569
- factor(KitchenQ)    3     31307 799903  61.658
- factor(GarageType)  5     74819 843415  88.483
- factor(ExteriorQ)   3     93092 861688 105.427
- factor(BasementHt)  3    106667 875263 115.044

Step:  AIC=42.75
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(KitchenQ) + 
    factor(ExteriorC) + factor(CentralAir) + factor(GarageQ) + 
    factor(BasementHt) + factor(GarageType) + factor(LotConfig) + 
    factor(BasementC) + factor(Condition)

                     Df Sum of Sq    RSS      Cp
- factor(ExteriorC)   3      4529 780565  39.959
- factor(GarageQ)     3      6360 782395  41.256
- factor(CentralAir)  1       770 776805  41.296
- factor(Condition)   7     18432 794467  41.808
<none>                            776035  42.750
- factor(BasementC)   3      9778 785813  43.677
- factor(LotConfig)   4     16348 792384  46.332
- factor(HouseStyle)  6     32643 808678  53.875
- factor(KitchenQ)    3     27594 803629  56.298
- factor(GarageType)  5     80955 856991  90.100
- factor(ExteriorQ)   3    101505 877541 108.658
- factor(BasementHt)  3    112313 888348 116.314

Step:  AIC=39.96
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(KitchenQ) + 
    factor(CentralAir) + factor(GarageQ) + factor(BasementHt) + 
    factor(GarageType) + factor(LotConfig) + factor(BasementC) + 
    factor(Condition)

                     Df Sum of Sq    RSS      Cp
- factor(Condition)   7     16942 797507  37.961
- factor(GarageQ)     3      5865 786430  38.114
- factor(CentralAir)  1       785 781349  38.515
<none>                            780565  39.959
- factor(BasementC)   3      8757 789322  40.163
- factor(LotConfig)   4     15985 796550  43.283
- factor(HouseStyle)  6     32949 813514  51.300
- factor(KitchenQ)    3     27491 808056  53.434
- factor(GarageType)  5     85120 865684  90.258
- factor(ExteriorQ)   3    100216 880781 104.953
- factor(BasementHt)  3    114195 894760 114.856

Step:  AIC=37.96
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(KitchenQ) + 
    factor(CentralAir) + factor(GarageQ) + factor(BasementHt) + 
    factor(GarageType) + factor(LotConfig) + factor(BasementC)

                     Df Sum of Sq    RSS      Cp
- factor(GarageQ)     3      6059 803566  36.254
- factor(CentralAir)  1      2686 800193  37.864
<none>                            797507  37.961
- factor(BasementC)   3     10393 807900  39.324
- factor(LotConfig)   4     15974 813482  41.278
- factor(HouseStyle)  6     34061 831568  50.090
- factor(KitchenQ)    3     31930 829437  54.581
- factor(GarageType)  5     84122 881629  87.554
- factor(ExteriorQ)   3    111475 908982 110.931
- factor(BasementHt)  3    115425 912932 113.729

Step:  AIC=36.25
Price ~ factor(HouseStyle) + factor(ExteriorQ) + factor(KitchenQ) + 
    factor(CentralAir) + factor(BasementHt) + factor(GarageType) + 
    factor(LotConfig) + factor(BasementC)

                     Df Sum of Sq    RSS      Cp
<none>                            803566  36.254
- factor(CentralAir)  1      3230 806796  36.542
- factor(BasementC)   3      9627 813193  37.073
- factor(LotConfig)   4     18042 821609  41.035
- factor(HouseStyle)  6     33772 837338  48.178
- factor(KitchenQ)    3     36568 840134  56.158
- factor(GarageType)  6     91186 894753  88.851
- factor(ExteriorQ)   3    110197 913763 108.318
- factor(BasementHt)  3    114370 917937 111.275

Call:
lm(formula = Price ~ factor(HouseStyle) + factor(ExteriorQ) + 
    factor(KitchenQ) + factor(CentralAir) + factor(BasementHt) + 
    factor(GarageType) + factor(LotConfig) + factor(BasementC), 
    data = AmesTrain6a)

Coefficients:
              (Intercept)   factor(HouseStyle)1.5Unf   factor(HouseStyle)1Story   factor(HouseStyle)2.5Unf  
                  405.314                     -8.381                     -1.405                     32.462  
 factor(HouseStyle)2Story   factor(HouseStyle)SFoyer     factor(HouseStyle)SLvl        factor(ExteriorQ)Fa  
                   13.102                    -21.945                     -5.033                   -130.500  
      factor(ExteriorQ)Gd        factor(ExteriorQ)TA         factor(KitchenQ)Fa         factor(KitchenQ)Gd  
                  -66.651                   -100.787                    -55.427                    -33.385  
       factor(KitchenQ)TA        factor(CentralAir)Y       factor(BasementHt)Fa       factor(BasementHt)Gd  
                  -44.764                     10.535                    -98.775                    -60.523  
   factor(BasementHt)None       factor(BasementHt)TA   factor(GarageType)Attchd  factor(GarageType)Basment  
                 -110.859                    -76.305                    -19.688                    -38.752  
factor(GarageType)BuiltIn  factor(GarageType)CarPort   factor(GarageType)Detchd     factor(GarageType)None  
                  -12.034                    -69.540                    -50.137                    -50.481  
 factor(LotConfig)CulDSac       factor(LotConfig)FR2       factor(LotConfig)FR3    factor(LotConfig)Inside  
                    9.261                    -20.069                     13.020                     -7.452  
      factor(BasementC)Fa        factor(BasementC)Gd      factor(BasementC)None        factor(BasementC)TA  
                  -12.318                     -0.857                         NA                    -20.085  
modCatReduced = lm (Price ~ factor(ExteriorQ) + factor(BasementHt) + 
    factor(GarageType) + factor(KitchenQ) + factor(HouseStyle) + 
    factor(LotConfig) + factor(Condition), data = AmesTrain6a)
modCatFull=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
Cp455(modCatReduced, modCatFull)
[1] 36.36975

After doing forward, backward, and stepwise regression, we’re going to use the categorical models that forward and stepwise gave us because of the low AIC and Mallow Cp values. The AIC for forward and stepwise was 36.37 and the AIC for backward selection was 36.25. Although the backward selection AIC was slightly lower, we chose to use the factors suggested by forward and stepwise selection because they agreed, and also because we may do a second round of selection when we combine these variables with the numerical variables. This means that we will include ExteriorQ, BasementHt, GarageType, KitchenQ, HouseStyle, LotConfig, and Condition in the next model. Additionally, the CP for the model suggested by stepwise selection is 36.369, which is substantially more than the number of variables in the model, which means the model isn’t very efficient. With that said, this CP is about the same between forward, backward, and stepwise selection, and may improve when we include numerical variables. Furthermore, a higher CP makes sense for when this many variables are introduced into a model.

Transformations of predictors.

We chose these predictors to transform based on the stepwise, backward, and forward selection that we did for Assignment #3

modLotArea=lm(Price~LotArea, data=AmesTrain6a)
modLotAreaSquared=lm(Price~LotArea+I(LotArea^2), data=AmesTrain6a)
modLotAreaSqrt=lm(Price~LotArea+I(sqrt(LotArea)), data=AmesTrain6a)
modLotAreaLog=lm(Price~(log(LotArea)), data=AmesTrain6a)
modLotAreaAll=lm(Price~LotArea+I(LotArea^2)+I(sqrt(LotArea))+I(log(LotArea)), data=AmesTrain6a)
anova(modLotArea, modLotAreaSquared, modLotAreaSqrt, modLotAreaLog, modLotAreaAll)
Analysis of Variance Table

Model 1: Price ~ LotArea
Model 2: Price ~ LotArea + I(LotArea^2)
Model 3: Price ~ LotArea + I(sqrt(LotArea))
Model 4: Price ~ (log(LotArea))
Model 5: Price ~ LotArea + I(LotArea^2) + I(sqrt(LotArea)) + I(log(LotArea))
  Res.Df     RSS Df Sum of Sq       F    Pr(>F)    
1    593 2602283                                   
2    592 2523892  1     78391 18.9784 1.559e-05 ***
3    592 2567183  0    -43291                      
4    593 2582422 -1    -15238  3.6892   0.05525 .  
5    590 2437001  3    145420 11.7355 1.776e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modLotArea$residuals~modLotArea$fitted.values)
abline(0,0)

plot(modLotAreaSquared$residuals~modLotAreaSquared$fitted.values)
abline(0,0)

This data shows that we should use LotArea^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.

modYearBuilt=lm(Price~YearBuilt, data=AmesTrain6a)
modYearBuiltSquared=lm(Price~YearBuilt+I(YearBuilt^2), data=AmesTrain6a)
modYearBuiltSqrt=lm(Price~YearBuilt+I(sqrt(YearBuilt)), data=AmesTrain6a)
modYearBuiltLog=lm(Price~(log(YearBuilt)), data=AmesTrain6a)
modYearBuiltFull=lm(Price)~YearBuilt+I(YearBuilt^2)+I(sqrt(YearBuilt))+I(log(YearBuilt), data=AmesTrain6a)
anova(modYearBuilt, modYearBuiltSquared, modYearBuiltSqrt, modYearBuiltLog, modYearBuiltFull)
models with response ‘"NULL"’ removed because response differs from model 1
Analysis of Variance Table

Model 1: Price ~ YearBuilt
Model 2: Price ~ YearBuilt + I(YearBuilt^2)
Model 3: Price ~ YearBuilt + I(sqrt(YearBuilt))
Model 4: Price ~ (log(YearBuilt))
  Res.Df     RSS Df Sum of Sq       F    Pr(>F)    
1    593 1896928                                   
2    592 1624934  1    271994  99.094 < 2.2e-16 ***
3    592 1626098  0     -1165                      
4    593 1906324 -1   -280226 102.093 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modYearBuilt$residuals~modYearBuilt$fitted.values)
abline(0,0)

plot(modYearBuiltSquared$residuals~modYearBuiltSquared$fitted.values)
abline(0,0)

This shows that we should use YearBuilt^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.

modWoodDeckSF=lm(Price~WoodDeckSF, data=AmesTrain6a)
modWoodDeckSFSquared=lm(Price~WoodDeckSF+I(WoodDeckSF^2), data=AmesTrain6a)
modWoodDeckSFSqrt=lm(Price~WoodDeckSF+I(sqrt(WoodDeckSF)), data=AmesTrain6a)
modWoodDeckSFLog=lm(Price~(log(WoodDeckSF+1)), data=AmesTrain6a)
modWoodDeckSFFull=lm(Price)~WoodDeckSF+I(WoodDeckSF^2)+I(sqrt(WoodDeckSF))+I(log(WoodDeckSF+1), data=AmesTrain6a)
anova(modWoodDeckSF, modWoodDeckSFSquared, modWoodDeckSFSqrt, modWoodDeckSFLog, modWoodDeckSFFull)
models with response ‘"NULL"’ removed because response differs from model 1
Analysis of Variance Table

Model 1: Price ~ WoodDeckSF
Model 2: Price ~ WoodDeckSF + I(WoodDeckSF^2)
Model 3: Price ~ WoodDeckSF + I(sqrt(WoodDeckSF))
Model 4: Price ~ (log(WoodDeckSF + 1))
  Res.Df     RSS Df Sum of Sq       F   Pr(>F)    
1    593 2542776                                  
2    592 2485078  1     57698 13.7450 0.000229 ***
3    592 2498904  0    -13825                     
4    593 2521822 -1    -22918  5.4596 0.019794 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modWoodDeckSF$residuals~modWoodDeckSF$fitted.values)
abline(0,0)

plot(modWoodDeckSFSquared$residuals~modWoodDeckSFSquared$fitted.values)
abline(0,0)

This shows that we should use WoodDeck^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.

modGroundSF=lm(Price~GroundSF, data=AmesTrain6a)
modGroundSFSquared=lm(Price~GroundSF+I(GroundSF^2), data=AmesTrain6a)
modGroundSFSqrt=lm(Price~GroundSF+I(sqrt(GroundSF)), data=AmesTrain6a)
modGroundSFLog=lm(Price~(log(GroundSF+1)), data=AmesTrain6a)
modGroundSFFull=lm(Price)~GroundSFF+I(GroundSF^2)+I(sqrt(GroundSF))+I(log(GroundSF+1), data=AmesTrain6a)
anova(modGroundSF, modGroundSFSquared, modGroundSFSqrt, modGroundSFLog, modGroundSFFull)
models with response ‘"NULL"’ removed because response differs from model 1
Analysis of Variance Table

Model 1: Price ~ GroundSF
Model 2: Price ~ GroundSF + I(GroundSF^2)
Model 3: Price ~ GroundSF + I(sqrt(GroundSF))
Model 4: Price ~ (log(GroundSF + 1))
  Res.Df     RSS Df Sum of Sq       F    Pr(>F)    
1    593 1465476                                   
2    592 1462787  1      2689  1.0884    0.2973    
3    592 1463302  0      -515                      
4    593 1541522 -1    -78221 31.6564 2.841e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modGroundSF$residuals~modGroundSF$fitted.values)
abline(0,0)

plot(modGroundSFSquared$residuals~modGroundSFSquared$fitted.values)
abline(0,0)

We’re keeping GroundSF as our transformation because it has the best balance between the number of variables used and a significant p-value.

modFullBath = lm(Price~FullBath, data = AmesTrain6a)
modFullBathSquared= lm(Price ~ FullBath+I(FullBath^2), data = AmesTrain6a)
modFullBathSqrt= lm(Price~FullBath+I(sqrt(FullBath)), data=AmesTrain6a)
modFullBathLog= lm(Price~(log(FullBath+1)), data=AmesTrain6a)
modFullBathFull = lm(Price~FullBath+I(FullBath^2)+I(sqrt(FullBath))+I(log(FullBath+1)), data=AmesTrain6a)
anova(modFullBath, modFullBathSquared, modFullBathSqrt, modFullBathLog, modFullBathFull)
Analysis of Variance Table

Model 1: Price ~ FullBath
Model 2: Price ~ FullBath + I(FullBath^2)
Model 3: Price ~ FullBath + I(sqrt(FullBath))
Model 4: Price ~ (log(FullBath + 1))
Model 5: Price ~ FullBath + I(FullBath^2) + I(sqrt(FullBath)) + I(log(FullBath + 
    1))
  Res.Df     RSS Df Sum of Sq       F   Pr(>F)   
1    593 1982921                                 
2    592 1982652  1       269  0.0808 0.776323   
3    592 1973813  0      8839                    
4    593 2007246 -1    -33433 10.0419 0.001609 **
5    591 1967617  2     39628  5.9514 0.002761 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modFullBath$residuals~modFullBath$fitted.values)
abline(0,0)

plot(modFullBathSquared$residuals~modFullBathSquared$fitted.values)
abline(0,0)

This shows that we should use FullBath^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.

modTotalRooms = lm(Price ~ TotalRooms, data = AmesTrain6a)
modTotalRoomsSquared = lm(Price ~ TotalRooms+I(TotalRooms^2), data = AmesTrain6a)
modTotalRoomsSqrt = lm(Price~TotalRooms+I(sqrt(TotalRooms)), data=AmesTrain6a)
modTotalRoomsLog = lm(Price~(log(TotalRooms+1)), data=AmesTrain6a)
modTotalRoomsFull = lm(Price~TotalRooms+I(TotalRooms^2)+I(sqrt(TotalRooms))+I(log(TotalRooms+1)), data=AmesTrain6a)
anova(modTotalRooms, modTotalRoomsSquared, modTotalRoomsSqrt, modTotalRoomsLog, modTotalRoomsFull)
Analysis of Variance Table

Model 1: Price ~ TotalRooms
Model 2: Price ~ TotalRooms + I(TotalRooms^2)
Model 3: Price ~ TotalRooms + I(sqrt(TotalRooms))
Model 4: Price ~ (log(TotalRooms + 1))
Model 5: Price ~ TotalRooms + I(TotalRooms^2) + I(sqrt(TotalRooms)) + 
    I(log(TotalRooms + 1))
  Res.Df     RSS Df Sum of Sq      F  Pr(>F)  
1    593 2280320                              
2    592 2280004  1     315.2 0.0823 0.77434  
3    592 2280299  0    -294.6                 
4    593 2291422 -1  -11123.3 2.9036 0.08891 .
5    590 2260256  3   31166.9 2.7119 0.04425 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modTotalRooms$residuals~modTotalRooms$fitted.values)
abline(0,0)

plot(modTotalRoomsSquared$residuals~modTotalRoomsSquared$fitted.values)
abline(0,0)

This shows that we should use TotalRooms + I(TotalRooms^2) + I(sqrt(TotalRooms)) + I(log(TotalRooms + 1)) as our transformation because it is the only one with a signficant p-value. We may not include this variable in a final model because of how many variables it creates and relies on.

modBasementSF=lm(Price~BasementSF, data=AmesTrain6a)
modBasementSFSquared=lm(Price~BasementSF+I(BasementSF^2), data=AmesTrain6a)
modBasementSFSqrt=lm(Price~BasementSF+I(sqrt(BasementSF)), data=AmesTrain6a)
modBasementSFLog=lm(Price~(log(BasementSF+1)), data=AmesTrain6a)
modBasementSFFull=lm(Price~BasementSF+I(BasementSF^2)+I(sqrt(BasementSF))+I(log(BasementSF+1)), data=AmesTrain6a)
anova(modBasementSF, modBasementSFSquared, modBasementSFSqrt, modBasementSFLog, modBasementSFFull)
Analysis of Variance Table

Model 1: Price ~ BasementSF
Model 2: Price ~ BasementSF + I(BasementSF^2)
Model 3: Price ~ BasementSF + I(sqrt(BasementSF))
Model 4: Price ~ (log(BasementSF + 1))
Model 5: Price ~ BasementSF + I(BasementSF^2) + I(sqrt(BasementSF)) + 
    I(log(BasementSF + 1))
  Res.Df     RSS Df Sum of Sq       F    Pr(>F)    
1    593 1882672                                   
2    592 1783033  1     99639  33.025 1.459e-08 ***
3    592 1800490  0    -17457                      
4    593 2661798 -1   -861308 285.477 < 2.2e-16 ***
5    590 1780081  3    881717  97.414 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modBasementSF$residuals~modBasementSF$fitted.values)
abline(0,0)

plot(modBasementSFSquared$residuals~modBasementSFSquared$fitted.values)
abline(0,0)

This shows that we should use log(BasementSF) as our transformation because it has the best balance between the number of variables used and a significant p-value.

modGarageCars=lm(Price~GarageCars, data=AmesTrain6a)
modGarageCarsSquared=lm(Price~GarageCars+I(GarageCars^2), data=AmesTrain6a)
modGarageCarsSqrt=lm(Price~GarageCars+I(sqrt(GarageCars)), data=AmesTrain6a)
modGarageCarsLog=lm(Price~(log(GarageCars+1)), data=AmesTrain6a)
modGarageCarsFull=lm(Price~GarageCars+I(GarageCars^2)+I(sqrt(GarageCars))+I(log(GarageCars+1)), data=AmesTrain6a)
anova(modGarageCars, modGarageCarsSquared, modGarageCarsSqrt, modGarageCarsLog, modGarageCarsFull)
Analysis of Variance Table

Model 1: Price ~ GarageCars
Model 2: Price ~ GarageCars + I(GarageCars^2)
Model 3: Price ~ GarageCars + I(sqrt(GarageCars))
Model 4: Price ~ (log(GarageCars + 1))
Model 5: Price ~ GarageCars + I(GarageCars^2) + I(sqrt(GarageCars)) + 
    I(log(GarageCars + 1))
  Res.Df     RSS Df Sum of Sq       F    Pr(>F)    
1    593 1774895                                   
2    592 1612893  1    162002  60.288 3.639e-14 ***
3    592 1650724  0    -37831                      
4    593 2004694 -1   -353970 131.728 < 2.2e-16 ***
5    590 1585410  3    419284  52.011 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(modGarageCars$residuals~modGarageCars$fitted.values)
abline(0,0)

plot(modGarageCarsSquared$residuals~modGarageCarsSquared$fitted.values)
abline(0,0)

This shows that we should use modGarageCars^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.

Transformations of the Response

modTransformNumericLog=lm(log(Price)~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms, data=AmesTrain6a)
summary(modTransformNumericLog)
modTransformNumeric=lm(Price~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms, data=AmesTrain6a)
summary(modTransformNumeric)
modTransformCatLog=lm(log(Price)~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCatLog)
modTransformCat=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCat)

We decided not to log the response because the results of the logged Price were not significantly different. Although the adjusted r-squared was slightly better (.001) for log(Price) than Price with numeric variables, Price was significantly better than log(Price) for Categorical variables (.02). Additionally, we didn’t want to overfit the data through too many transformations, so we chose to keep Price as the response variable.

Combinations of Variables

modAllBathroom=lm(Price~FullBath+BasementFBath+0.5*BasementHBath+0.5*HalfBath, data-AmesTrain6a)
summary(modAllBathroom)

We chose not to combine any of the variables because they didn’t significantly improve the model. For example, experimental combinations with the different bath variables didn’t improve the adjusted r-squared value while also lowering AIC and Mallow Cp.

Final Selection for Fancy Model

modTransformCat=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCat)

Call:
lm(formula = Price ~ factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(HeatingQC) + factor(KitchenQ) + 
    factor(ExteriorC) + factor(CentralAir) + factor(GarageQ) + 
    factor(Foundation) + factor(GarageC) + factor(BasementHt) + 
    factor(GarageType) + factor(LotConfig) + factor(BasementC) + 
    factor(Heating) + factor(Condition), data = AmesTrain6a)

Residuals:
    Min      1Q  Median      3Q     Max 
-96.316 -23.460  -0.895  18.138 154.613 

Coefficients: (4 not defined because of singularities)
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)                276.8311    87.8232   3.152 0.001712 ** 
factor(HouseStyle)1.5Unf    -4.5273    20.4211  -0.222 0.824633    
factor(HouseStyle)1Story    -0.0779     6.5346  -0.012 0.990493    
factor(HouseStyle)2.5Unf    42.7315    23.8827   1.789 0.074150 .  
factor(HouseStyle)2Story    11.6310     7.0351   1.653 0.098862 .  
factor(HouseStyle)SFoyer   -24.9542    12.1385  -2.056 0.040291 *  
factor(HouseStyle)SLvl      -7.8712     9.2634  -0.850 0.395873    
factor(ExteriorQ)Fa       -127.2075    23.2606  -5.469 6.99e-08 ***
factor(ExteriorQ)Gd        -62.8435    13.3102  -4.721 3.00e-06 ***
factor(ExteriorQ)TA        -94.3985    14.1932  -6.651 7.27e-11 ***
factor(BasementFin)BLQ       0.8396     7.0710   0.119 0.905529    
factor(BasementFin)GLQ       3.5429     5.5164   0.642 0.520991    
factor(BasementFin)LwQ       1.7849     8.8851   0.201 0.840869    
factor(BasementFin)None   -110.4109    42.1347  -2.620 0.009034 ** 
factor(BasementFin)Rec       1.2895     7.3736   0.175 0.861240    
factor(BasementFin)Unf      -6.0051     5.3489  -1.123 0.262085    
factor(HeatingQC)Fa         -0.6798    11.2329  -0.061 0.951765    
factor(HeatingQC)Gd          4.0802     5.3941   0.756 0.449737    
factor(HeatingQC)TA         -1.1773     4.9969  -0.236 0.813823    
factor(KitchenQ)Fa         -52.7203    15.1922  -3.470 0.000562 ***
factor(KitchenQ)Gd         -34.8167     9.5749  -3.636 0.000304 ***
factor(KitchenQ)TA         -40.5129    10.1530  -3.990 7.53e-05 ***
factor(ExteriorC)Fa          4.3272    31.9069   0.136 0.892173    
factor(ExteriorC)Gd        -25.8265    26.4996  -0.975 0.330203    
factor(ExteriorC)TA        -19.6775    27.3093  -0.721 0.471509    
factor(CentralAir)Y          5.2874     8.6428   0.612 0.540953    
factor(GarageQ)Gd           61.5677    27.1955   2.264 0.023984 *  
factor(GarageQ)None         46.7622    50.8032   0.920 0.357753    
factor(GarageQ)Po           49.5994    70.6183   0.702 0.482764    
factor(GarageQ)TA            0.1246    10.8491   0.011 0.990844    
factor(Foundation)CBlock     1.7521     7.6006   0.231 0.817772    
factor(Foundation)PConc     11.4451     8.6758   1.319 0.187676    
factor(Foundation)Slab      12.2294    22.4476   0.545 0.586120    
factor(Foundation)Stone     18.3872    24.4244   0.753 0.451891    
factor(Foundation)Wood      24.3737    29.0463   0.839 0.401774    
factor(GarageC)Fa           85.3064    48.2085   1.770 0.077380 .  
factor(GarageC)Gd           98.6830    50.8363   1.941 0.052765 .  
factor(GarageC)None              NA         NA      NA       NA    
factor(GarageC)Po           88.2301    55.9473   1.577 0.115387    
factor(GarageC)TA           98.0079    46.5230   2.107 0.035617 *  
factor(BasementHt)Fa       -89.7863    14.3966  -6.237 9.14e-10 ***
factor(BasementHt)Gd       -58.4549     8.2455  -7.089 4.33e-12 ***
factor(BasementHt)None           NA         NA      NA       NA    
factor(BasementHt)TA       -72.2207    10.0079  -7.216 1.86e-12 ***
factor(GarageType)Attchd   -17.1134    18.0483  -0.948 0.343460    
factor(GarageType)Basment  -32.2733    23.4157  -1.378 0.168700    
factor(GarageType)BuiltIn   -5.6891    19.0984  -0.298 0.765907    
factor(GarageType)CarPort  -58.3085    45.1565  -1.291 0.197178    
factor(GarageType)Detchd   -46.0371    18.1132  -2.542 0.011317 *  
factor(GarageType)None           NA         NA      NA       NA    
factor(LotConfig)CulDSac     7.4004     7.6958   0.962 0.336682    
factor(LotConfig)FR2       -20.1713     8.8796  -2.272 0.023509 *  
factor(LotConfig)FR3        10.9377    20.5572   0.532 0.594905    
factor(LotConfig)Inside     -7.3204     4.6136  -1.587 0.113172    
factor(BasementC)Fa         -7.9727    39.4513  -0.202 0.839924    
factor(BasementC)Gd         -1.8171    39.2841  -0.046 0.963124    
factor(BasementC)None            NA         NA      NA       NA    
factor(BasementC)TA        -18.5301    38.3507  -0.483 0.629172    
factor(Heating)GasW         24.5236    17.0449   1.439 0.150806    
factor(Heating)Grav        -43.9947    45.3212  -0.971 0.332125    
factor(Heating)OthW         -1.8212    41.8501  -0.044 0.965306    
factor(Heating)Wall          8.0581    32.3359   0.249 0.803302    
factor(Condition)3           0.4042    51.1716   0.008 0.993701    
factor(Condition)4          22.0054    50.2147   0.438 0.661400    
factor(Condition)5          35.7435    50.0151   0.715 0.475137    
factor(Condition)6          33.5797    50.1292   0.670 0.503237    
factor(Condition)7          36.8567    50.2968   0.733 0.464014    
factor(Condition)8          44.5512    50.5200   0.882 0.378256    
factor(Condition)9          49.7814    52.3594   0.951 0.342158    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 37.57 on 530 degrees of freedom
Multiple R-squared:  0.7414,    Adjusted R-squared:  0.7102 
F-statistic: 23.75 on 64 and 530 DF,  p-value: < 2.2e-16
MSE=(summary(modTransformCat)$sigma)^2
step(none,scope=list(upper=modTransformCat),scale=MSE)
Start:  AIC=1456.74
Price ~ 1

                      Df Sum of Sq     RSS      Cp
+ factor(ExteriorQ)    3   1606904 1286545  324.40
+ factor(BasementHt)   4   1555913 1337536  362.52
+ factor(KitchenQ)     3   1305178 1588271  538.14
+ factor(Foundation)   5    973531 1919917  777.09
+ factor(GarageType)   6    737589 2155859  946.23
+ factor(HeatingQC)    3    635156 2258293 1012.79
+ factor(Condition)    7    546078 2347371 1083.90
+ factor(BasementFin)  6    515220 2378228 1103.76
+ factor(HouseStyle)   6    230783 2662666 1305.26
+ factor(GarageC)      5    220424 2673025 1310.59
+ factor(GarageQ)      4    213737 2679712 1313.33
+ factor(CentralAir)   1    171099 2722350 1337.54
+ factor(BasementC)    4    116360 2777088 1382.31
+ factor(ExteriorC)    3     73758 2819691 1410.49
+ factor(LotConfig)    4     40148 2853301 1436.30
+ factor(Heating)      4     31951 2861498 1442.11
<none>                             2893449 1456.74

Step:  AIC=324.4
Price ~ factor(ExteriorQ)

                      Df Sum of Sq     RSS      Cp
+ factor(BasementHt)   4    250784 1035761  154.74
+ factor(GarageType)   6    213599 1072946  185.08
+ factor(Foundation)   5    127908 1158637  243.79
+ factor(KitchenQ)     3    111799 1174746  251.20
+ factor(GarageC)      5     61878 1224668  290.57
+ factor(CentralAir)   1     49963 1236582  291.01
+ factor(GarageQ)      4     51506 1235039  295.91
+ factor(Condition)    7     56831 1229715  298.14
+ factor(HouseStyle)   6     51999 1234546  299.56
+ factor(BasementFin)  6     50313 1236232  300.76
+ factor(HeatingQC)    3     34665 1251880  305.84
+ factor(LotConfig)    4     37324 1249222  305.96
+ factor(BasementC)    4     31824 1254721  309.86
+ factor(Heating)      4     14719 1271826  321.97
+ factor(ExteriorC)    3      9120 1277425  323.94
<none>                             1286545  324.40
- factor(ExteriorQ)    3   1606904 2893449 1456.74

Step:  AIC=154.74
Price ~ factor(ExteriorQ) + factor(BasementHt)

                      Df Sum of Sq     RSS      Cp
+ factor(GarageType)   6    124591  911170  78.481
+ factor(KitchenQ)     3     58853  976908 119.050
+ factor(GarageQ)      4     38749  997012 135.292
+ factor(HouseStyle)   6     43956  991805 135.604
+ factor(GarageC)      5     38839  996922 137.228
+ factor(CentralAir)   1     23287 1012474 140.246
+ factor(Foundation)   5     32902 1002859 141.434
+ factor(LotConfig)    4     24518 1011243 145.374
+ factor(HeatingQC)    3     19358 1016403 147.029
+ factor(Condition)    7     29957 1005804 147.521
<none>                             1035761 154.742
+ factor(ExteriorC)    3      6387 1029374 156.217
+ factor(Heating)      4      9177 1026584 156.241
+ factor(BasementC)    3      5603 1030159 156.773
+ factor(BasementFin)  5      9762 1025999 157.827
- factor(BasementHt)   4    250784 1286545 324.400
- factor(ExteriorQ)    3    301775 1337536 362.522

Step:  AIC=78.48
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType)

                      Df Sum of Sq     RSS      Cp
+ factor(KitchenQ)     3     46077  865093  51.839
+ factor(HouseStyle)   6     33520  877650  66.735
+ factor(Condition)    7     30314  880856  71.006
+ factor(LotConfig)    4     17289  893881  74.233
+ factor(GarageQ)      3     13232  897938  75.107
+ factor(BasementC)    3     11289  899881  76.484
+ factor(CentralAir)   1      4507  906662  77.288
+ factor(HeatingQC)    3      9175  901995  77.981
+ factor(ExteriorC)    3      8888  902282  78.184
+ factor(Foundation)   5     14356  896814  78.311
<none>                              911170  78.481
+ factor(GarageC)      4      7043  904127  81.491
+ factor(Heating)      4      5425  905744  82.637
+ factor(BasementFin)  5      6296  904874  84.021
- factor(GarageType)   6    124591 1035761 154.742
- factor(BasementHt)   4    161777 1072946 185.085
- factor(ExteriorQ)    3    262738 1173908 258.607

Step:  AIC=51.84
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ)

                      Df Sum of Sq    RSS      Cp
+ factor(HouseStyle)   6     29516 835576  42.930
+ factor(LotConfig)    4     16523 848569  48.134
+ factor(Condition)    7     20942 844151  51.004
+ factor(BasementC)    3      9422 855671  51.165
<none>                             865093  51.839
+ factor(CentralAir)   1      2384 862708  52.150
+ factor(GarageQ)      3      7667 857426  52.408
+ factor(Foundation)   5     11629 853463  53.601
+ factor(ExteriorC)    3      4888 860204  54.376
+ factor(GarageC)      4      7585 857508  54.466
+ factor(HeatingQC)    3      2954 862139  55.747
+ factor(Heating)      4      3609 861484  57.283
+ factor(BasementFin)  5      4771 860322  58.459
- factor(KitchenQ)     3     46077 911170  78.481
- factor(GarageType)   6    111815 976908 119.050
- factor(ExteriorQ)    3    111470 976562 124.805
- factor(BasementHt)   4    131934 997027 137.303

Step:  AIC=42.93
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle)

                      Df Sum of Sq    RSS      Cp
+ factor(LotConfig)    4     19231 816346  37.306
+ factor(BasementC)    3     10072 825504  41.795
+ factor(Condition)    7     21361 814215  41.797
+ factor(CentralAir)   1      3577 832000  42.396
<none>                             835576  42.930
+ factor(GarageQ)      3      7821 827756  43.389
+ factor(GarageC)      4      7300 828276  45.758
+ factor(ExteriorC)    3      3909 831667  46.160
+ factor(BasementFin)  5      8833 826743  46.672
+ factor(Foundation)   5      8749 826827  46.732
+ factor(HeatingQC)    3      2709 832867  47.010
+ factor(Heating)      4      2844 832732  48.915
- factor(HouseStyle)   6     29516 865093  51.839
- factor(KitchenQ)     3     42073 877650  66.735
- factor(GarageType)   6    103598 939174 104.319
- factor(ExteriorQ)    3    103129 938705 109.987
- factor(BasementHt)   4    135911 971487 131.210

Step:  AIC=37.31
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle) + factor(LotConfig)

                      Df Sum of Sq    RSS      Cp
+ factor(Condition)    7     21085 795261  36.370
+ factor(BasementC)    3      9549 806796  36.542
+ factor(CentralAir)   1      3152 813193  37.073
<none>                             816346  37.306
+ factor(GarageQ)      3      5790 810556  39.205
+ factor(BasementFin)  5      9685 806661  40.446
+ factor(ExteriorC)    3      3928 812418  40.524
+ factor(Foundation)   5      9138 807207  40.833
+ factor(GarageC)      4      6012 810334  41.048
+ factor(HeatingQC)    3      2690 813656  41.401
- factor(LotConfig)    4     19231 835576  42.930
+ factor(Heating)      4      2312 814033  43.669
- factor(HouseStyle)   6     32224 848569  48.134
- factor(KitchenQ)     3     41042 857388  60.381
- factor(GarageType)   6     98041 914387  94.760
- factor(ExteriorQ)    3    107192 923538 107.243
- factor(BasementHt)   4    128373 944719 120.247

Step:  AIC=36.37
Price ~ factor(ExteriorQ) + factor(BasementHt) + factor(GarageType) + 
    factor(KitchenQ) + factor(HouseStyle) + factor(LotConfig) + 
    factor(Condition)

                      Df Sum of Sq    RSS      Cp
<none>                             795261  36.370
+ factor(BasementC)    3      7759 787502  36.873
- factor(Condition)    7     21085 816346  37.306
+ factor(CentralAir)   1       825 794436  37.786
+ factor(GarageQ)      3      5350 789911  38.580
+ factor(Foundation)   5     10768 784493  38.742
+ factor(ExteriorC)    3      3198 792063  40.104
+ factor(BasementFin)  5      8249 787012  40.526
+ factor(HeatingQC)    3      1677 793584  41.182
+ factor(GarageC)      4      3844 791417  41.647
- factor(LotConfig)    4     18954 814215  41.797
+ factor(Heating)      4      2858 792403  42.345
- factor(HouseStyle)   6     32660 827921  47.506
- factor(KitchenQ)     3     33199 828460  53.889
- factor(GarageType)   6     98676 893937  94.273
- factor(ExteriorQ)    3     97091 892352  99.150
- factor(BasementHt)   4    126197 921458 117.769

Call:
lm(formula = Price ~ factor(ExteriorQ) + factor(BasementHt) + 
    factor(GarageType) + factor(KitchenQ) + factor(HouseStyle) + 
    factor(LotConfig) + factor(Condition), data = AmesTrain6a)

Coefficients:
              (Intercept)        factor(ExteriorQ)Fa        factor(ExteriorQ)Gd        factor(ExteriorQ)TA  
                  371.587                   -127.471                    -64.446                    -97.138  
     factor(BasementHt)Fa       factor(BasementHt)Gd     factor(BasementHt)None       factor(BasementHt)TA  
                 -101.217                    -61.932                    -93.392                    -80.750  
 factor(GarageType)Attchd  factor(GarageType)Basment  factor(GarageType)BuiltIn  factor(GarageType)CarPort  
                  -15.095                    -33.555                     -7.098                    -67.326  
 factor(GarageType)Detchd     factor(GarageType)None         factor(KitchenQ)Fa         factor(KitchenQ)Gd  
                  -46.666                    -47.290                    -55.587                    -34.705  
       factor(KitchenQ)TA   factor(HouseStyle)1.5Unf   factor(HouseStyle)1Story   factor(HouseStyle)2.5Unf  
                  -43.160                    -12.327                     -2.912                     32.865  
 factor(HouseStyle)2Story   factor(HouseStyle)SFoyer     factor(HouseStyle)SLvl   factor(LotConfig)CulDSac  
                    9.627                    -28.005                     -8.037                      9.395  
     factor(LotConfig)FR2       factor(LotConfig)FR3    factor(LotConfig)Inside         factor(Condition)3  
                  -20.856                     12.616                     -7.700                     -6.741  
       factor(Condition)4         factor(Condition)5         factor(Condition)6         factor(Condition)7  
                    7.656                     22.628                     20.582                     24.883  
       factor(Condition)8         factor(Condition)9  
                   32.250                     41.650  
modTransformNumeric=lm(Price~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms, data=AmesTrain6a)
summary(modTransformNumeric)

Call:
lm(formula = Price ~ LotArea + I(LotArea^2) + YearBuilt + I(YearBuilt^2) + 
    BasementSF + I(BasementSF^2) + GarageCars + I(GarageCars^2) + 
    WoodDeckSF + I(WoodDeckSF^2) + GroundSF + I(GroundSF^2) + 
    FullBath + TotalRooms, data = AmesTrain6a)

Residuals:
     Min       1Q   Median       3Q      Max 
-119.888  -17.804   -0.971   14.951  140.922 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)      2.871e+04  5.613e+03   5.114 4.29e-07 ***
LotArea          4.700e-03  1.016e-03   4.626 4.59e-06 ***
I(LotArea^2)    -1.361e-07  3.983e-08  -3.418 0.000675 ***
YearBuilt       -3.001e+01  5.740e+00  -5.229 2.38e-07 ***
I(YearBuilt^2)   7.858e-03  1.468e-03   5.355 1.24e-07 ***
BasementSF      -1.685e-03  1.199e-02  -0.140 0.888336    
I(BasementSF^2)  1.845e-05  5.519e-06   3.343 0.000881 ***
GarageCars      -9.694e+00  6.511e+00  -1.489 0.137091    
I(GarageCars^2)  5.723e+00  1.974e+00   2.898 0.003893 ** 
WoodDeckSF       1.197e-01  2.794e-02   4.282 2.17e-05 ***
I(WoodDeckSF^2) -2.188e-04  7.421e-05  -2.949 0.003320 ** 
GroundSF         3.191e-02  1.748e-02   1.826 0.068412 .  
I(GroundSF^2)    1.557e-05  4.724e-06   3.296 0.001041 ** 
FullBath        -1.051e+01  3.621e+00  -2.901 0.003859 ** 
TotalRooms      -5.953e+00  1.600e+00  -3.721 0.000218 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 31.81 on 580 degrees of freedom
Multiple R-squared:  0.7972,    Adjusted R-squared:  0.7923 
F-statistic: 162.8 on 14 and 580 DF,  p-value: < 2.2e-16
MSE=(summary(modTransformNumeric)$sigma)^2
step(none,scope=list(upper=modTransformNumeric),scale=MSE)
Start:  AIC=2266.82
Price ~ 1

                  Df Sum of Sq     RSS      Cp
+ GroundSF         1   1427972 1465476  857.44
+ I(GroundSF^2)    1   1394445 1499004  890.58
+ I(GarageCars^2)  1   1276249 1617200 1007.40
+ GarageCars       1   1118553 1774895 1163.27
+ I(BasementSF^2)  1   1109930 1783519 1171.79
+ BasementSF       1   1010777 1882672 1269.79
+ I(YearBuilt^2)   1   1005760 1887689 1274.75
+ YearBuilt        1    996521 1896928 1283.88
+ FullBath         1    910528 1982921 1368.87
+ TotalRooms       1    613129 2280320 1662.82
+ WoodDeckSF       1    350672 2542776 1922.22
+ LotArea          1    291166 2602283 1981.04
+ I(WoodDeckSF^2)  1    185418 2708031 2085.56
+ I(LotArea^2)     1    169321 2724127 2101.47
<none>                         2893449 2266.82

Step:  AIC=857.44
Price ~ GroundSF

                  Df Sum of Sq     RSS      Cp
+ I(YearBuilt^2)   1    532682  932795  332.95
+ YearBuilt        1    530339  935137  335.27
+ I(BasementSF^2)  1    391579 1073898  472.42
+ BasementSF       1    377033 1088444  486.79
+ I(GarageCars^2)  1    339234 1126243  524.15
+ GarageCars       1    285885 1179592  576.88
+ WoodDeckSF       1    113090 1352386  747.67
+ TotalRooms       1     90961 1374516  769.54
+ FullBath         1     57487 1407989  802.62
+ I(WoodDeckSF^2)  1     35416 1430060  824.44
+ LotArea          1     16082 1449394  843.55
+ I(LotArea^2)     1      2964 1462513  856.51
+ I(GroundSF^2)    1      2689 1462787  856.79
<none>                         1465476  857.44
- GroundSF         1   1427972 2893449 2266.82

Step:  AIC=332.95
Price ~ GroundSF + I(YearBuilt^2)

                  Df Sum of Sq     RSS      Cp
+ I(BasementSF^2)  1    183645  749150  153.44
+ BasementSF       1    170574  762221  166.36
+ I(GarageCars^2)  1     89747  843047  246.25
+ GarageCars       1     48072  884723  287.44
+ WoodDeckSF       1     43893  888902  291.57
+ YearBuilt        1     33500  899295  301.84
+ LotArea          1     30861  901934  304.45
+ I(WoodDeckSF^2)  1     23620  909175  311.61
+ TotalRooms       1     22576  910219  312.64
+ I(GroundSF^2)    1     13999  918796  321.12
+ I(LotArea^2)     1     12374  920421  322.72
+ FullBath         1      9402  923393  325.66
<none>                          932795  332.95
- I(YearBuilt^2)   1    532682 1465476  857.44
- GroundSF         1    954894 1887689 1274.75

Step:  AIC=153.44
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2)

                  Df Sum of Sq     RSS     Cp
+ I(GarageCars^2)  1     40765  708385 115.15
+ YearBuilt        1     35776  713374 120.08
+ WoodDeckSF       1     25372  723778 130.37
+ GarageCars       1     22075  727075 133.62
+ I(GroundSF^2)    1     19629  729521 136.04
+ TotalRooms       1     13510  735640 142.09
+ LotArea          1     10237  738913 145.33
+ FullBath         1      8302  740849 147.24
+ I(WoodDeckSF^2)  1      8291  740859 147.25
+ I(LotArea^2)     1      3740  745410 151.75
<none>                          749150 153.44
+ BasementSF       1       361  748789 155.09
- I(BasementSF^2)  1    183645  932795 332.95
- I(YearBuilt^2)   1    324748 1073898 472.42
- GroundSF         1    590645 1339795 735.22

Step:  AIC=115.15
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2)

                  Df Sum of Sq     RSS      Cp
+ WoodDeckSF       1     21590  686794  95.812
+ YearBuilt        1     21252  687133  96.147
+ I(GroundSF^2)    1     16503  691882 100.840
+ TotalRooms       1     14971  693414 102.355
+ FullBath         1     10282  698103 106.989
+ GarageCars       1      9688  698697 107.577
+ LotArea          1      8453  699932 108.797
+ I(WoodDeckSF^2)  1      7416  700969 109.822
+ I(LotArea^2)     1      2759  705626 114.425
<none>                          708385 115.152
+ BasementSF       1       900  707484 116.262
- I(GarageCars^2)  1     40765  749150 153.443
- I(BasementSF^2)  1    134663  843047 246.249
- I(YearBuilt^2)   1    209182  917567 319.902
- GroundSF         1    380195 1088580 488.928

Step:  AIC=95.81
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF

                  Df Sum of Sq     RSS      Cp
+ YearBuilt        1     21448  665346  76.613
+ I(GroundSF^2)    1     14411  672384  83.569
+ TotalRooms       1     13645  673149  84.326
+ I(WoodDeckSF^2)  1     11895  674899  86.055
+ GarageCars       1      9410  677385  88.512
+ FullBath         1      9346  677449  88.575
+ LotArea          1      7069  679725  90.826
+ I(LotArea^2)     1      2216  684578  95.622
<none>                          686794  95.812
+ BasementSF       1       951  685843  96.872
- WoodDeckSF       1     21590  708385 115.152
- I(GarageCars^2)  1     36984  723778 130.366
- I(BasementSF^2)  1    122584  809379 214.972
- I(YearBuilt^2)   1    192934  879729 284.504
- GroundSF         1    361179 1047973 450.794

Step:  AIC=76.61
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt

                  Df Sum of Sq    RSS      Cp
+ I(GroundSF^2)    1     19727 645619  59.116
+ FullBath         1     18008 647338  60.815
+ TotalRooms       1     13928 651419  64.848
+ LotArea          1     11641 653705  67.108
+ I(WoodDeckSF^2)  1      7143 658204  71.554
+ GarageCars       1      5119 660227  73.554
+ I(LotArea^2)     1      4682 660664  73.985
<none>                         665346  76.613
+ BasementSF       1       886 664460  77.737
- YearBuilt        1     21448 686794  95.812
- WoodDeckSF       1     21787 687133  96.147
- I(YearBuilt^2)   1     22682 688029  97.032
- I(GarageCars^2)  1     23268 688615  97.611
- I(BasementSF^2)  1    129396 794742 202.506
- GroundSF         1    305148 970494 376.215

Step:  AIC=59.12
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2)

                  Df Sum of Sq    RSS      Cp
+ LotArea          1     14148 631471  47.132
+ FullBath         1     11777 633843  49.476
+ TotalRooms       1     10290 635329  50.945
+ I(WoodDeckSF^2)  1      8145 637474  53.065
+ I(LotArea^2)     1      5908 639711  55.276
- GroundSF         1         6 645625  57.121
<none>                         645619  59.116
+ GarageCars       1      1400 644219  59.732
+ BasementSF       1       621 644998  60.502
- WoodDeckSF       1     19353 664973  76.244
- I(GarageCars^2)  1     19622 665241  76.509
- I(GroundSF^2)    1     19727 665346  76.613
- YearBuilt        1     26765 672384  83.569
- I(YearBuilt^2)   1     28164 673783  84.953
- I(BasementSF^2)  1    136647 782266 192.174

Step:  AIC=47.13
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2) + LotArea

                  Df Sum of Sq    RSS      Cp
+ FullBath         1     11468 620002  37.797
+ TotalRooms       1     10759 620711  38.497
+ I(WoodDeckSF^2)  1      9942 621528  39.305
+ I(LotArea^2)     1      8913 622558  40.322
- GroundSF         1       368 631839  45.496
<none>                         631471  47.132
+ GarageCars       1      1735 629735  47.416
+ BasementSF       1       883 630588  48.259
- LotArea          1     14148 645619  59.116
- I(GarageCars^2)  1     16765 648236  61.702
- WoodDeckSF       1     17392 648863  62.322
- I(GroundSF^2)    1     22235 653705  67.108
- YearBuilt        1     32676 664147  77.428
- I(YearBuilt^2)   1     34242 665713  78.976
- I(BasementSF^2)  1    121959 753429 165.673

Step:  AIC=37.8
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2) + LotArea + FullBath

                  Df Sum of Sq    RSS      Cp
+ I(WoodDeckSF^2)  1      9818 610184  30.093
+ I(LotArea^2)     1      9362 610640  30.543
+ TotalRooms       1      8165 611837  31.726
- GroundSF         1       281 620283  36.074
<none>                         620002  37.797
+ GarageCars       1      1773 618229  38.044
+ BasementSF       1       601 619402  39.203
- FullBath         1     11468 631471  47.132
- LotArea          1     13840 633843  49.476
- I(GroundSF^2)    1     15599 635602  51.215
- WoodDeckSF       1     16815 636817  52.416
- I(GarageCars^2)  1     17166 637169  52.763
- YearBuilt        1     39547 659549  74.884
- I(YearBuilt^2)   1     41333 661335  76.649
- I(BasementSF^2)  1    120447 740449 154.844

Step:  AIC=30.09
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2) + LotArea + FullBath + 
    I(WoodDeckSF^2)

                  Df Sum of Sq    RSS      Cp
+ TotalRooms       1      8914 601270  23.282
+ I(LotArea^2)     1      7997 602188  24.189
- GroundSF         1       203 610388  28.294
<none>                         610184  30.093
+ GarageCars       1      1555 608629  30.555
+ BasementSF       1       271 609914  31.825
- I(WoodDeckSF^2)  1      9818 620002  37.797
- FullBath         1     11344 621528  39.305
- LotArea          1     15609 625793  43.520
- I(GarageCars^2)  1     16074 626258  43.980
- I(GroundSF^2)    1     16735 626919  44.633
- WoodDeckSF       1     20936 631120  48.785
- YearBuilt        1     32824 643008  60.535
- I(YearBuilt^2)   1     34342 644526  62.036
- I(BasementSF^2)  1    125712 735897 152.344

Step:  AIC=23.28
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2) + LotArea + FullBath + 
    I(WoodDeckSF^2) + TotalRooms

                  Df Sum of Sq    RSS      Cp
+ I(LotArea^2)     1     12165 589105  13.259
+ GarageCars       1      2384 598886  22.926
- GroundSF         1      1920 603191  23.180
<none>                         601270  23.282
+ BasementSF       1       188 601082  25.097
- FullBath         1      8659 609930  29.841
- TotalRooms       1      8914 610184  30.093
- I(WoodDeckSF^2)  1     10567 611837  31.726
- I(GroundSF^2)    1     14417 615687  35.531
- LotArea          1     16168 617439  37.263
- I(GarageCars^2)  1     16908 618178  37.993
- WoodDeckSF       1     21577 622847  42.608
- YearBuilt        1     31490 632761  52.407
- I(YearBuilt^2)   1     32894 634164  53.794
- I(BasementSF^2)  1    119088 720358 138.986

Step:  AIC=13.26
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2) + LotArea + FullBath + 
    I(WoodDeckSF^2) + TotalRooms + I(LotArea^2)

                  Df Sum of Sq    RSS      Cp
+ GarageCars       1      2265 586840  13.020
<none>                         589105  13.259
- GroundSF         1      2281 591386  13.513
+ BasementSF       1        43 589063  15.216
- FullBath         1      8572 597677  19.731
- I(WoodDeckSF^2)  1      8989 598094  20.143
- I(LotArea^2)     1     12165 601270  23.282
- TotalRooms       1     13082 602188  24.189
- I(GroundSF^2)    1     14593 603698  25.681
- I(GarageCars^2)  1     17119 606225  28.179
- WoodDeckSF       1     18817 607922  29.856
- LotArea          1     21975 611080  32.978
- YearBuilt        1     32298 621403  43.181
- I(YearBuilt^2)   1     33713 622818  44.580
- I(BasementSF^2)  1    105176 694281 115.212

Step:  AIC=13.02
Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + I(GarageCars^2) + 
    WoodDeckSF + YearBuilt + I(GroundSF^2) + LotArea + FullBath + 
    I(WoodDeckSF^2) + TotalRooms + I(LotArea^2) + GarageCars

                  Df Sum of Sq    RSS      Cp
<none>                         586840  13.020
- GarageCars       1      2265 589105  13.259
- GroundSF         1      3385 590225  14.365
+ BasementSF       1        20 586820  15.000
- FullBath         1      8496 595336  19.417
- I(GarageCars^2)  1      8580 595421  19.500
- I(WoodDeckSF^2)  1      8783 595623  19.700
- I(GroundSF^2)    1     10972 597812  21.864
- I(LotArea^2)     1     12046 598886  22.926
- TotalRooms       1     13997 600837  24.854
- WoodDeckSF       1     18542 605382  29.346
- LotArea          1     22005 608845  32.769
- YearBuilt        1     27649 614489  38.347
- I(YearBuilt^2)   1     28994 615834  39.677
- I(BasementSF^2)  1     95658 682498 105.566

Call:
lm(formula = Price ~ GroundSF + I(YearBuilt^2) + I(BasementSF^2) + 
    I(GarageCars^2) + WoodDeckSF + YearBuilt + I(GroundSF^2) + 
    LotArea + FullBath + I(WoodDeckSF^2) + TotalRooms + I(LotArea^2) + 
    GarageCars, data = AmesTrain6a)

Coefficients:
    (Intercept)         GroundSF   I(YearBuilt^2)  I(BasementSF^2)  I(GarageCars^2)       WoodDeckSF  
      2.869e+04        3.196e-02        7.855e-03        1.772e-05        5.739e+00        1.194e-01  
      YearBuilt    I(GroundSF^2)          LotArea         FullBath  I(WoodDeckSF^2)       TotalRooms  
     -3.000e+01        1.555e-05        4.677e-03       -1.048e+01       -2.181e-04       -5.939e+00  
   I(LotArea^2)       GarageCars  
     -1.351e-07       -9.734e+00  

We chose to narrow down our pool of variables separately by categorical and numerical factors before we combined them. It was easier to analyze the numeric and categorical variables in models together. However, once we narrowed down the categorical and numerical variables seperately, we combined them in this model (modTransformFull) and re-ran stepwise, forward, and backward selection, which is below.

modTransformFull=lm(Price~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms+factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(KitchenQ)+factor(BasementHt)+factor(Condition), data=AmesTrain6a)
summary(modTransformFull)

Call:
lm(formula = Price ~ LotArea + I(LotArea^2) + YearBuilt + I(YearBuilt^2) + 
    BasementSF + I(BasementSF^2) + GarageCars + I(GarageCars^2) + 
    WoodDeckSF + I(WoodDeckSF^2) + GroundSF + I(GroundSF^2) + 
    FullBath + TotalRooms + factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(KitchenQ) + factor(BasementHt) + 
    factor(Condition), data = AmesTrain6a)

Residuals:
   Min     1Q Median     3Q    Max 
-86.54 -12.42  -1.22  11.65 112.04 

Coefficients: (1 not defined because of singularities)
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)               6.353e+03  5.703e+03   1.114 0.265747    
LotArea                   4.730e-03  7.867e-04   6.012 3.34e-09 ***
I(LotArea^2)             -1.443e-07  3.053e-08  -4.727 2.89e-06 ***
YearBuilt                -6.891e+00  5.857e+00  -1.176 0.239934    
I(YearBuilt^2)            1.896e-03  1.504e-03   1.261 0.207799    
BasementSF                9.301e-03  1.548e-02   0.601 0.548100    
I(BasementSF^2)           2.151e-06  6.334e-06   0.340 0.734324    
GarageCars               -6.668e+00  4.962e+00  -1.344 0.179609    
I(GarageCars^2)           4.167e+00  1.499e+00   2.779 0.005632 ** 
WoodDeckSF                3.723e-02  2.138e-02   1.742 0.082123 .  
I(WoodDeckSF^2)          -7.028e-05  5.609e-05  -1.253 0.210778    
GroundSF                  5.170e-02  1.458e-02   3.546 0.000424 ***
I(GroundSF^2)             8.937e-06  3.653e-06   2.447 0.014734 *  
FullBath                 -7.836e+00  2.773e+00  -2.825 0.004892 ** 
TotalRooms               -3.449e+00  1.250e+00  -2.760 0.005979 ** 
factor(HouseStyle)1.5Unf  2.978e+01  1.161e+01   2.565 0.010581 *  
factor(HouseStyle)1Story  1.274e+01  4.706e+00   2.707 0.007007 ** 
factor(HouseStyle)2.5Unf  1.059e+01  1.468e+01   0.721 0.470962    
factor(HouseStyle)2Story  4.084e+00  4.376e+00   0.933 0.351062    
factor(HouseStyle)SFoyer  1.662e+00  7.493e+00   0.222 0.824604    
factor(HouseStyle)SLvl    8.772e+00  5.608e+00   1.564 0.118315    
factor(ExteriorQ)Fa      -7.441e+01  1.354e+01  -5.497 5.92e-08 ***
factor(ExteriorQ)Gd      -4.390e+01  8.287e+00  -5.297 1.70e-07 ***
factor(ExteriorQ)TA      -6.023e+01  8.985e+00  -6.704 5.03e-11 ***
factor(BasementFin)BLQ    6.407e-01  4.208e+00   0.152 0.879041    
factor(BasementFin)GLQ    3.279e+00  3.373e+00   0.972 0.331435    
factor(BasementFin)LwQ   -9.849e+00  5.290e+00  -1.862 0.063155 .  
factor(BasementFin)None  -5.252e+01  1.285e+01  -4.087 5.02e-05 ***
factor(BasementFin)Rec   -6.342e-01  4.504e+00  -0.141 0.888090    
factor(BasementFin)Unf   -8.310e+00  3.269e+00  -2.542 0.011286 *  
factor(KitchenQ)Fa       -2.945e+01  8.783e+00  -3.353 0.000855 ***
factor(KitchenQ)Gd       -2.078e+01  5.712e+00  -3.638 0.000301 ***
factor(KitchenQ)TA       -2.097e+01  5.906e+00  -3.550 0.000418 ***
factor(BasementHt)Fa     -4.652e+01  8.935e+00  -5.207 2.71e-07 ***
factor(BasementHt)Gd     -3.494e+01  5.209e+00  -6.707 4.93e-11 ***
factor(BasementHt)None           NA         NA      NA       NA    
factor(BasementHt)TA     -4.414e+01  6.429e+00  -6.866 1.79e-11 ***
factor(Condition)3        3.319e+00  2.715e+01   0.122 0.902732    
factor(Condition)4        1.302e+01  2.654e+01   0.491 0.623871    
factor(Condition)5        2.482e+01  2.631e+01   0.943 0.346075    
factor(Condition)6        3.357e+01  2.636e+01   1.274 0.203352    
factor(Condition)7        4.118e+01  2.636e+01   1.562 0.118857    
factor(Condition)8        4.995e+01  2.643e+01   1.890 0.059297 .  
factor(Condition)9        6.333e+01  2.722e+01   2.326 0.020363 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 23.41 on 552 degrees of freedom
Multiple R-squared:  0.8954,    Adjusted R-squared:  0.8875 
F-statistic: 112.5 on 42 and 552 DF,  p-value: < 2.2e-16

We made this model by combining the categorical and numerical variables suggested by our separate stepwise selections.

Stepwise selection:

MSE=(summary(modTransformFull)$sigma)^2
step(none,scope=list(upper=modTransformFull),scale=MSE)
Start:  AIC=4685.16
Price ~ 1

                      Df Sum of Sq     RSS     Cp
+ factor(ExteriorQ)    3   1606904 1286545 1759.9
+ factor(BasementHt)   4   1555913 1337536 1854.9
+ GroundSF             1   1427972 1465476 2082.3
+ I(GroundSF^2)        1   1394445 1499004 2143.4
+ factor(KitchenQ)     3   1305178 1588271 2310.3
+ I(GarageCars^2)      1   1276249 1617200 2359.1
+ GarageCars           1   1118553 1774895 2646.7
+ I(BasementSF^2)      1   1109930 1783519 2662.5
+ BasementSF           1   1010777 1882672 2843.3
+ I(YearBuilt^2)       1   1005760 1887689 2852.5
+ YearBuilt            1    996521 1896928 2869.3
+ FullBath             1    910528 1982921 3026.2
+ TotalRooms           1    613129 2280320 3568.7
+ factor(Condition)    7    546078 2347371 3703.0
+ factor(BasementFin)  6    515220 2378228 3757.3
+ WoodDeckSF           1    350672 2542776 4047.5
+ LotArea              1    291166 2602283 4156.0
+ factor(HouseStyle)   6    230783 2662666 4276.2
+ I(WoodDeckSF^2)      1    185418 2708031 4348.9
+ I(LotArea^2)         1    169321 2724127 4378.3
<none>                             2893449 4685.2

Step:  AIC=1759.89
Price ~ factor(ExteriorQ)

                      Df Sum of Sq     RSS      Cp
+ I(GroundSF^2)        1    559835  726710  740.65
+ GroundSF             1    551843  734702  755.23
+ I(GarageCars^2)      1    281632 1004913 1248.14
+ I(BasementSF^2)      1    265658 1020887 1277.28
+ BasementSF           1    253821 1032724 1298.87
+ TotalRooms           1    250074 1036472 1305.71
+ factor(BasementHt)   4    250784 1035761 1310.41
+ GarageCars           1    238478 1048067 1326.86
+ FullBath             1    202387 1084158 1392.70
+ LotArea              1    169637 1116908 1452.44
+ I(YearBuilt^2)       1    148483 1138062 1491.03
+ YearBuilt            1    147695 1138850 1492.46
+ factor(KitchenQ)     3    111799 1174746 1561.94
+ I(LotArea^2)         1    100624 1185921 1578.33
+ WoodDeckSF           1     86278 1200267 1604.50
+ I(WoodDeckSF^2)      1     71648 1214897 1631.19
+ factor(Condition)    7     56831 1229715 1670.22
+ factor(HouseStyle)   6     51999 1234546 1677.03
+ factor(BasementFin)  6     50313 1236232 1680.11
<none>                             1286545 1759.89
- factor(ExteriorQ)    3   1606904 2893449 4685.16

Step:  AIC=740.65
Price ~ factor(ExteriorQ) + I(GroundSF^2)

                      Df Sum of Sq     RSS      Cp
+ factor(BasementHt)   4    160720  565990  455.47
+ I(YearBuilt^2)       1    157397  569313  455.53
+ YearBuilt            1    157382  569329  455.56
+ BasementSF           1    121924  604786  520.24
+ I(BasementSF^2)      1    112159  614551  538.05
+ factor(HouseStyle)   6    113570  613140  545.48
+ factor(BasementFin)  6     92363  634347  584.16
+ I(GarageCars^2)      1     66215  660495  621.86
+ GarageCars           1     63555  663155  626.71
+ factor(KitchenQ)     3     56948  669762  642.77
+ factor(Condition)    7     53872  672838  656.38
+ WoodDeckSF           1     33869  692841  680.86
+ LotArea              1     30692  696018  686.66
+ I(WoodDeckSF^2)      1     16874  709836  711.87
+ TotalRooms           1     13037  713673  718.87
+ I(LotArea^2)         1     11312  715398  722.01
+ FullBath             1      6776  719934  730.29
+ GroundSF             1      3391  723319  736.46
<none>                              726710  740.65
- I(GroundSF^2)        1    559835 1286545 1759.89
- factor(ExteriorQ)    3    772294 1499004 2143.45

Step:  AIC=455.47
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt)

                      Df Sum of Sq     RSS      Cp
+ factor(HouseStyle)   6     84078  481912  314.09
+ BasementSF           1     74839  491150  320.95
+ YearBuilt            1     66808  499182  335.60
+ I(YearBuilt^2)       1     66606  499384  335.96
+ I(BasementSF^2)      1     65947  500043  337.17
+ GarageCars           1     37702  528288  388.69
+ I(GarageCars^2)      1     37418  528572  389.21
+ factor(BasementFin)  5     39757  526233  392.94
+ LotArea              1     34700  531290  394.17
+ factor(Condition)    7     37467  528522  401.12
+ factor(KitchenQ)     3     29273  536717  408.07
+ WoodDeckSF           1     14324  551666  431.34
+ I(LotArea^2)         1     13501  552489  432.84
+ I(WoodDeckSF^2)      1     10439  555551  438.42
+ TotalRooms           1      5224  560766  447.94
+ GroundSF             1      3983  562007  450.20
<none>                              565990  455.47
+ FullBath             1       200  565790  457.10
- factor(BasementHt)   4    160720  726710  740.65
- factor(ExteriorQ)    3    169432  735422  758.54
- I(GroundSF^2)        1    469771 1035761 1310.41

Step:  AIC=314.09
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle)

                      Df Sum of Sq    RSS      Cp
+ factor(Condition)    7     38427 443485  257.99
+ YearBuilt            1     30128 451783  261.13
+ I(YearBuilt^2)       1     30095 451816  261.19
+ factor(BasementFin)  5     26815 455097  275.18
+ factor(KitchenQ)     3     24506 457406  275.39
+ GarageCars           1     20826 461085  278.10
+ I(GarageCars^2)      1     20337 461574  278.99
+ BasementSF           1     19221 462690  281.03
+ I(BasementSF^2)      1     13778 468134  290.96
+ GroundSF             1     12741 469171  292.85
+ LotArea              1     12573 469338  293.16
+ WoodDeckSF           1      8474 473438  300.63
+ I(WoodDeckSF^2)      1      4728 477184  307.47
+ I(LotArea^2)         1      2753 479159  311.07
<none>                             481912  314.09
+ TotalRooms           1      1025 480887  314.22
+ FullBath             1       125 481787  315.86
- factor(HouseStyle)   6     84078 565990  455.47
- factor(BasementHt)   4    131228 613140  545.48
- factor(ExteriorQ)    3    136831 618743  557.70
- I(GroundSF^2)        1    509894 991805 1242.23

Step:  AIC=257.99
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition)

                      Df Sum of Sq    RSS      Cp
+ I(YearBuilt^2)       1     47529 395956  173.29
+ YearBuilt            1     47420 396065  173.49
+ BasementSF           1     25980 417505  212.60
+ I(GarageCars^2)      1     21214 422271  221.30
+ GarageCars           1     21029 422456  221.63
+ I(BasementSF^2)      1     19924 423561  223.65
+ factor(BasementFin)  5     20154 423331  231.23
+ factor(KitchenQ)     3     16659 426826  233.61
+ LotArea              1     14369 429116  233.78
+ GroundSF             1     13497 429988  235.37
+ WoodDeckSF           1      5209 438276  250.49
+ I(LotArea^2)         1      3787 439697  253.09
+ I(WoodDeckSF^2)      1      2155 441329  256.06
<none>                             443485  257.99
+ TotalRooms           1       437 443048  259.20
+ FullBath             1        97 443388  259.82
- factor(Condition)    7     38427 481912  314.09
- factor(HouseStyle)   6     85038 528522  401.12
- factor(ExteriorQ)    3    119422 562907  469.84
- factor(BasementHt)   4    127941 571426  483.38
- I(GroundSF^2)        1    521124 964609 1206.62

Step:  AIC=173.29
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2)

                      Df Sum of Sq    RSS      Cp
+ BasementSF           1     19141 376814  140.38
+ LotArea              1     17163 378793  143.99
+ I(BasementSF^2)      1     15213 380743  147.54
+ I(GarageCars^2)      1     14269 381686  149.26
+ GarageCars           1     11136 384819  154.98
+ GroundSF             1     10966 384990  155.29
+ factor(KitchenQ)     3     11718 384237  157.92
+ factor(BasementFin)  5     13149 382807  159.31
+ I(LotArea^2)         1      6413 389543  163.59
+ WoodDeckSF           1      5133 390822  165.93
+ I(WoodDeckSF^2)      1      2604 393352  170.54
+ FullBath             1      1859 394097  171.90
<none>                             395956  173.29
+ TotalRooms           1       648 395307  174.11
+ YearBuilt            1       403 395553  174.56
- factor(HouseStyle)   6     48883 444838  250.46
- I(YearBuilt^2)       1     47529 443485  257.99
- factor(Condition)    7     55861 451816  261.19
- factor(BasementHt)   4     62522 458478  279.34
- factor(ExteriorQ)    3     91958 487914  335.04
- I(GroundSF^2)        1    520628 916584 1121.01

Step:  AIC=140.38
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF

                      Df Sum of Sq    RSS     Cp
+ LotArea              1     15952 360862 113.28
+ I(GarageCars^2)      1      9570 367245 124.92
+ factor(KitchenQ)     3     11622 365192 125.18
+ GarageCars           1      7546 369268 128.61
+ factor(BasementFin)  5     11477 365337 129.44
+ I(LotArea^2)         1      6477 370337 130.56
+ WoodDeckSF           1      3870 372945 135.32
+ FullBath             1      3450 373364 136.08
+ GroundSF             1      3147 373667 136.63
+ I(WoodDeckSF^2)      1      1520 375294 139.60
+ TotalRooms           1      1403 375411 139.82
<none>                             376814 140.38
+ I(BasementSF^2)      1       579 376236 141.32
+ YearBuilt            1       325 376490 141.78
- factor(HouseStyle)   6      7915 384730 142.81
- BasementSF           1     19141 395956 173.29
- I(YearBuilt^2)       1     40691 417505 212.60
- factor(BasementHt)   4     44791 421605 214.08
- factor(Condition)    7     62009 438823 239.49
- factor(ExteriorQ)    3     79015 455829 278.51
- I(GroundSF^2)        1    224483 601297 547.87

Step:  AIC=113.28
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea

                      Df Sum of Sq    RSS      Cp
+ I(LotArea^2)         1     10708 350155  95.744
+ factor(KitchenQ)     3     12063 348800  97.272
+ I(GarageCars^2)      1      7903 352959 100.860
+ factor(BasementFin)  5     11063 349799 103.096
+ GarageCars           1      5588 355274 105.083
+ FullBath             1      3607 357255 108.696
- factor(HouseStyle)   6      4395 365257 109.293
+ WoodDeckSF           1      3146 357716 109.538
+ TotalRooms           1      2411 358452 110.879
<none>                             360862 113.277
+ GroundSF             1       903 359960 113.630
+ I(WoodDeckSF^2)      1       871 359991 113.688
+ YearBuilt            1       593 360269 114.195
+ I(BasementSF^2)      1       354 360508 114.631
- LotArea              1     15952 376814 140.376
- BasementSF           1     17931 378793 143.986
- factor(BasementHt)   4     43851 404713 185.268
- I(YearBuilt^2)       1     43356 404218 190.365
- factor(Condition)    7     62571 423434 213.418
- factor(ExteriorQ)    3     81904 442767 256.684
- I(GroundSF^2)        1    167305 528168 416.471

Step:  AIC=95.74
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2)

                      Df Sum of Sq    RSS      Cp
+ factor(KitchenQ)     3     11542 338612  80.689
+ factor(BasementFin)  5     12613 337542  82.736
+ I(GarageCars^2)      1      7747 342408  83.612
+ GarageCars           1      5656 344499  87.427
+ TotalRooms           1      4897 345258  88.811
+ FullBath             1      4220 345935  90.046
- factor(HouseStyle)   6      4599 354753  92.133
+ WoodDeckSF           1      2703 347451  92.812
<none>                             350155  95.744
+ I(WoodDeckSF^2)      1       896 349258  96.109
+ GroundSF             1       697 349458  96.473
+ YearBuilt            1       373 349782  97.064
+ I(BasementSF^2)      1         4 350151  97.737
- I(LotArea^2)         1     10708 360862 113.277
- BasementSF           1     15192 365346 121.457
- LotArea              1     20183 370337 130.561
- I(YearBuilt^2)       1     39181 389336 165.217
- factor(BasementHt)   4     46395 396549 172.376
- factor(Condition)    7     61003 411157 193.024
- factor(ExteriorQ)    3     82504 432659 240.246
- I(GroundSF^2)        1    163484 513639 391.968

Step:  AIC=80.69
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ)

                      Df Sum of Sq    RSS      Cp
+ I(GarageCars^2)      1      7655 330957  68.724
+ factor(BasementFin)  5     11288 327324  70.097
+ GarageCars           1      5848 332764  72.021
+ TotalRooms           1      4510 334102  74.462
+ FullBath             1      4214 334399  75.002
- factor(HouseStyle)   6      4119 342731  76.203
+ WoodDeckSF           1      2159 336454  78.751
<none>                             338612  80.689
+ GroundSF             1       833 337779  81.169
+ I(WoodDeckSF^2)      1       823 337789  81.188
+ YearBuilt            1       107 338505  82.494
+ I(BasementSF^2)      1         4 338609  82.682
- factor(KitchenQ)     3     11542 350155  95.744
- I(LotArea^2)         1     10187 348800  97.272
- BasementSF           1     15072 353684 106.183
- LotArea              1     19681 358293 114.591
- I(YearBuilt^2)       1     35073 373685 142.668
- factor(BasementHt)   4     39626 378239 144.974
- factor(ExteriorQ)    3     41745 380357 150.838
- factor(Condition)    7     51298 389911 160.266
- I(GroundSF^2)        1    155237 493849 361.868

Step:  AIC=68.72
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2)

                      Df Sum of Sq    RSS      Cp
+ factor(BasementFin)  5     13552 317405  54.002
+ TotalRooms           1      5177 325780  61.281
+ FullBath             1      4755 326202  62.051
- factor(HouseStyle)   6      4240 335197  64.459
+ WoodDeckSF           1      1838 329119  67.371
<none>                             330957  68.724
+ GroundSF             1       919 330038  69.048
+ I(WoodDeckSF^2)      1       730 330227  69.392
+ GarageCars           1       139 330818  70.470
+ I(BasementSF^2)      1        42 330915  70.647
+ YearBuilt            1         7 330950  70.711
- I(GarageCars^2)      1      7655 338612  80.689
- factor(KitchenQ)     3     11451 342408  83.612
- I(LotArea^2)         1     10064 341021  85.082
- BasementSF           1     11355 342312  87.438
- LotArea              1     18895 349853 101.193
- I(YearBuilt^2)       1     31854 362812 124.833
- factor(ExteriorQ)    3     36151 367108 128.670
- factor(BasementHt)   4     38327 369284 130.640
- factor(Condition)    7     51993 382950 149.569
- I(GroundSF^2)        1    136711 467669 316.110

Step:  AIC=54
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin)

                      Df Sum of Sq    RSS      Cp
- factor(HouseStyle)   6      4662 322066  50.506
+ FullBath             1      3012 314393  50.508
+ TotalRooms           1      2505 314899  51.432
+ GroundSF             1      1837 315568  52.652
+ WoodDeckSF           1      1111 316293  53.975
<none>                             317405  54.002
+ YearBuilt            1       486 316919  55.116
+ GarageCars           1       443 316962  55.194
+ I(WoodDeckSF^2)      1       297 317108  55.461
+ I(BasementSF^2)      1       111 317293  55.799
- factor(KitchenQ)     3     10333 327738  66.853
- factor(BasementFin)  5     13552 330957  68.724
- BasementSF           1      9243 326648  68.863
- I(GarageCars^2)      1      9919 327324  70.097
- I(LotArea^2)         1     11150 328555  72.343
- LotArea              1     19862 337266  88.234
- I(YearBuilt^2)       1     26724 344129 100.752
- factor(BasementHt)   3     34121 351526 110.245
- factor(ExteriorQ)    3     35972 353377 113.622
- factor(Condition)    7     44738 362142 121.612
- I(GroundSF^2)        1    139309 456714 306.127

Step:  AIC=50.51
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin)

                      Df Sum of Sq    RSS      Cp
+ TotalRooms           1      2875 319191  47.261
+ FullBath             1      2819 319247  47.363
+ WoodDeckSF           1      1114 320952  50.473
<none>                             322066  50.506
+ GroundSF             1       689 321377  51.249
+ YearBuilt            1       391 321675  51.792
+ I(WoodDeckSF^2)      1       315 321751  51.931
+ GarageCars           1       259 321807  52.034
+ I(BasementSF^2)      1        59 322007  52.398
+ factor(HouseStyle)   6      4662 317405  54.002
- factor(KitchenQ)     3     10432 332498  63.535
- factor(BasementFin)  5     13131 335197  64.459
- I(GarageCars^2)      1      9543 331610  65.915
- I(LotArea^2)         1     10966 333032  68.510
- LotArea              1     20710 342777  86.286
- BasementSF           1     25906 347972  95.762
- factor(BasementHt)   3     33072 355138 104.835
- I(YearBuilt^2)       1     32353 354419 107.523
- factor(ExteriorQ)    3     36352 358418 110.819
- factor(Condition)    7     46123 368189 120.642
- I(GroundSF^2)        1    245999 568065 497.251

Step:  AIC=47.26
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms

                      Df Sum of Sq    RSS      Cp
+ GroundSF             1      2642 316549  44.441
+ FullBath             1      1700 317491  46.160
<none>                             319191  47.261
+ WoodDeckSF           1      1055 318136  47.336
+ YearBuilt            1       701 318490  47.982
+ GarageCars           1       254 318937  48.797
+ I(WoodDeckSF^2)      1       253 318938  48.799
+ I(BasementSF^2)      1        61 319130  49.149
- TotalRooms           1      2875 322066  50.506
+ factor(HouseStyle)   6      4292 314899  51.432
- factor(BasementFin)  5     10407 329598  56.246
- factor(KitchenQ)     3     10086 329277  59.660
- I(GarageCars^2)      1      9807 328998  63.150
- I(LotArea^2)         1     12793 331984  68.598
- LotArea              1     23031 342222  87.274
- BasementSF           1     24625 343816  90.181
- factor(BasementHt)   3     33636 352827 102.619
- I(YearBuilt^2)       1     31999 351190 103.632
- factor(ExteriorQ)    3     36175 355366 107.250
- factor(Condition)    7     44509 363700 114.453
- I(GroundSF^2)        1    146108 465299 311.788

Step:  AIC=44.44
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms + GroundSF

                      Df Sum of Sq    RSS      Cp
+ FullBath             1      3126 313423  40.738
+ WoodDeckSF           1      1131 315418  44.377
<none>                             316549  44.441
+ GarageCars           1       810 315739  44.963
+ YearBuilt            1       561 315987  45.417
+ I(WoodDeckSF^2)      1       310 316239  45.875
+ factor(HouseStyle)   6      5761 310787  45.931
+ I(BasementSF^2)      1        28 316521  46.390
- GroundSF             1      2642 319191  47.261
- TotalRooms           1      4829 321377  51.249
- factor(BasementFin)  5     10764 327313  54.077
- factor(KitchenQ)     3     10165 326714  56.983
- I(GroundSF^2)        1      8198 324747  57.396
- I(GarageCars^2)      1      9974 326523  60.635
- I(LotArea^2)         1     13354 329903  66.801
- BasementSF           1     21321 337870  81.334
- LotArea              1     23190 339738  84.743
- factor(BasementHt)   3     34226 350774 100.875
- I(YearBuilt^2)       1     32343 348892 101.440
- factor(ExteriorQ)    3     34851 351400 102.015
- factor(Condition)    7     44476 361025 111.573

Step:  AIC=40.74
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms + GroundSF + FullBath

                      Df Sum of Sq    RSS      Cp
+ factor(HouseStyle)   6      6596 306826  40.705
<none>                             313423  40.738
+ WoodDeckSF           1      1043 312379  40.835
+ YearBuilt            1       964 312458  40.979
+ GarageCars           1       830 312593  41.224
+ I(WoodDeckSF^2)      1       257 313165  42.269
+ I(BasementSF^2)      1        26 313397  42.691
- FullBath             1      3126 316549  44.441
- TotalRooms           1      3743 317165  45.565
- GroundSF             1      4069 317491  46.160
- factor(BasementFin)  5      9978 323401  48.940
- I(GroundSF^2)        1      6256 319678  50.150
- factor(KitchenQ)     3     10200 323622  53.344
- I(GarageCars^2)      1     10322 323744  57.567
- I(LotArea^2)         1     13377 326800  63.140
- BasementSF           1     22371 335793  79.546
- LotArea              1     22885 336307  80.484
- factor(ExteriorQ)    3     33897 347319  96.572
- factor(BasementHt)   3     35224 348646  98.993
- I(YearBuilt^2)       1     34895 348317 102.392
- factor(Condition)    7     44972 358394 108.775

Step:  AIC=40.7
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms + GroundSF + FullBath + factor(HouseStyle)

                      Df Sum of Sq    RSS      Cp
+ GarageCars           1      1347 305479  40.248
+ YearBuilt            1      1184 305642  40.544
+ WoodDeckSF           1      1115 305711  40.672
<none>                             306826  40.705
- factor(HouseStyle)   6      6596 313423  40.738
+ I(WoodDeckSF^2)      1       282 306544  42.191
+ I(BasementSF^2)      1        61 306765  42.593
- TotalRooms           1      3566 310392  45.209
- FullBath             1      3961 310787  45.931
- I(GroundSF^2)        1      4167 310993  46.306
- BasementSF           1      4489 311315  46.893
- GroundSF             1      6096 312922  49.825
- factor(BasementFin)  5     10552 317378  49.954
- factor(KitchenQ)     3     10141 316967  53.204
- I(GarageCars^2)      1     10914 317741  58.615
- I(LotArea^2)         1     13555 320381  63.431
- LotArea              1     21061 327887  77.124
- I(YearBuilt^2)       1     28577 335403  90.834
- factor(ExteriorQ)    3     33591 340417  95.981
- factor(BasementHt)   3     36372 343198 101.053
- factor(Condition)    7     42818 349644 104.813

Step:  AIC=40.25
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms + GroundSF + FullBath + factor(HouseStyle) + GarageCars

                      Df Sum of Sq    RSS      Cp
+ WoodDeckSF           1      1114 304366  40.217
<none>                             305479  40.248
+ YearBuilt            1       858 304622  40.684
- GarageCars           1      1347 306826  40.705
- factor(HouseStyle)   6      7113 312593  41.224
+ I(WoodDeckSF^2)      1       291 305189  41.718
+ I(BasementSF^2)      1        49 305431  42.159
- I(GroundSF^2)        1      3069 308548  43.846
- BasementSF           1      3838 309318  45.250
- TotalRooms           1      3924 309404  45.407
- FullBath             1      4020 309500  45.582
- I(GarageCars^2)      1      5331 310810  47.973
- factor(BasementFin)  5     11036 316516  50.381
- GroundSF             1      7039 312518  51.088
- factor(KitchenQ)     3      9791 315271  52.109
- I(LotArea^2)         1     13376 318855  62.648
- LotArea              1     21055 326535  76.657
- I(YearBuilt^2)       1     29912 335391  92.813
- factor(ExteriorQ)    3     33112 338592  94.651
- factor(BasementHt)   3     35133 340612  98.336
- factor(Condition)    7     43312 348791 105.257

Step:  AIC=40.22
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms + GroundSF + FullBath + factor(HouseStyle) + GarageCars + 
    WoodDeckSF

                      Df Sum of Sq    RSS      Cp
<none>                             304366  40.217
- WoodDeckSF           1      1114 305479  40.248
+ I(WoodDeckSF^2)      1       935 303431  40.511
+ YearBuilt            1       887 303479  40.599
- GarageCars           1      1346 305711  40.672
- factor(HouseStyle)   6      7179 311544  41.312
+ I(BasementSF^2)      1        17 304349  42.186
- I(GroundSF^2)        1      2870 307235  43.452
- BasementSF           1      3580 307946  44.747
- TotalRooms           1      3910 308275  45.349
- FullBath             1      3929 308295  45.384
- I(GarageCars^2)      1      5246 309611  47.786
- factor(BasementFin)  5     10450 314816  49.280
- GroundSF             1      7200 311566  51.351
- factor(KitchenQ)     3      9474 313840  51.499
- I(LotArea^2)         1     13021 317387  61.970
- LotArea              1     20431 324797  75.487
- I(YearBuilt^2)       1     30239 334604  93.377
- factor(ExteriorQ)    3     33011 337376  94.434
- factor(BasementHt)   3     34043 338408  96.317
- factor(Condition)    7     41713 346079 102.309

Call:
lm(formula = Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(Condition) + I(YearBuilt^2) + BasementSF + LotArea + 
    I(LotArea^2) + factor(KitchenQ) + I(GarageCars^2) + factor(BasementFin) + 
    TotalRooms + GroundSF + FullBath + factor(HouseStyle) + GarageCars + 
    WoodDeckSF, data = AmesTrain6a)

Coefficients:
             (Intercept)       factor(ExteriorQ)Fa       factor(ExteriorQ)Gd       factor(ExteriorQ)TA  
              -3.578e+02                -7.652e+01                -4.455e+01                -6.249e+01  
           I(GroundSF^2)      factor(BasementHt)Fa      factor(BasementHt)Gd    factor(BasementHt)None  
               8.318e-06                -4.839e+01                -3.652e+01                -5.351e+01  
    factor(BasementHt)TA        factor(Condition)3        factor(Condition)4        factor(Condition)5  
              -4.732e+01                 5.008e+00                 1.510e+01                 2.705e+01  
      factor(Condition)6        factor(Condition)7        factor(Condition)8        factor(Condition)9  
               3.544e+01                 4.263e+01                 5.135e+01                 6.339e+01  
          I(YearBuilt^2)                BasementSF                   LotArea              I(LotArea^2)  
               1.281e-04                 1.360e-02                 4.725e-03                -1.464e-07  
      factor(KitchenQ)Fa        factor(KitchenQ)Gd        factor(KitchenQ)TA           I(GarageCars^2)  
              -3.115e+01                -2.127e+01                -2.217e+01                 4.559e+00  
  factor(BasementFin)BLQ    factor(BasementFin)GLQ    factor(BasementFin)LwQ   factor(BasementFin)None  
               7.728e-01                 3.401e+00                -9.447e+00                        NA  
  factor(BasementFin)Rec    factor(BasementFin)Unf                TotalRooms                  GroundSF  
              -9.769e-01                -7.594e+00                -3.331e+00                 5.274e-02  
                FullBath  factor(HouseStyle)1.5Unf  factor(HouseStyle)1Story  factor(HouseStyle)2.5Unf  
              -7.362e+00                 2.897e+01                 1.281e+01                 1.156e+01  
factor(HouseStyle)2Story  factor(HouseStyle)SFoyer    factor(HouseStyle)SLvl                GarageCars  
               4.640e+00                 1.714e+00                 8.855e+00                -7.686e+00  
              WoodDeckSF  
               1.299e-02  

Forward selection

MSE=(summary(modTransformFull)$sigma)^2
none=lm(Price~1,data=AmesTrain6a)
step(none,scope=list(upper=modTransformFull),scale=MSE, direction = "forward")
Start:  AIC=4685.16
Price ~ 1

                      Df Sum of Sq     RSS     Cp
+ factor(ExteriorQ)    3   1606904 1286545 1759.9
+ factor(BasementHt)   4   1555913 1337536 1854.9
+ GroundSF             1   1427972 1465476 2082.3
+ I(GroundSF^2)        1   1394445 1499004 2143.4
+ factor(KitchenQ)     3   1305178 1588271 2310.3
+ I(GarageCars^2)      1   1276249 1617200 2359.1
+ GarageCars           1   1118553 1774895 2646.7
+ I(BasementSF^2)      1   1109930 1783519 2662.5
+ BasementSF           1   1010777 1882672 2843.3
+ I(YearBuilt^2)       1   1005760 1887689 2852.5
+ YearBuilt            1    996521 1896928 2869.3
+ FullBath             1    910528 1982921 3026.2
+ TotalRooms           1    613129 2280320 3568.7
+ factor(Condition)    7    546078 2347371 3703.0
+ factor(BasementFin)  6    515220 2378228 3757.3
+ WoodDeckSF           1    350672 2542776 4047.5
+ LotArea              1    291166 2602283 4156.0
+ factor(HouseStyle)   6    230783 2662666 4276.2
+ I(WoodDeckSF^2)      1    185418 2708031 4348.9
+ I(LotArea^2)         1    169321 2724127 4378.3
<none>                             2893449 4685.2

Step:  AIC=1759.89
Price ~ factor(ExteriorQ)

                      Df Sum of Sq     RSS      Cp
+ I(GroundSF^2)        1    559835  726710  740.65
+ GroundSF             1    551843  734702  755.23
+ I(GarageCars^2)      1    281632 1004913 1248.14
+ I(BasementSF^2)      1    265658 1020887 1277.28
+ BasementSF           1    253821 1032724 1298.87
+ TotalRooms           1    250074 1036472 1305.71
+ factor(BasementHt)   4    250784 1035761 1310.41
+ GarageCars           1    238478 1048067 1326.86
+ FullBath             1    202387 1084158 1392.70
+ LotArea              1    169637 1116908 1452.44
+ I(YearBuilt^2)       1    148483 1138062 1491.03
+ YearBuilt            1    147695 1138850 1492.46
+ factor(KitchenQ)     3    111799 1174746 1561.94
+ I(LotArea^2)         1    100624 1185921 1578.33
+ WoodDeckSF           1     86278 1200267 1604.50
+ I(WoodDeckSF^2)      1     71648 1214897 1631.19
+ factor(Condition)    7     56831 1229715 1670.22
+ factor(HouseStyle)   6     51999 1234546 1677.03
+ factor(BasementFin)  6     50313 1236232 1680.11
<none>                             1286545 1759.89

Step:  AIC=740.65
Price ~ factor(ExteriorQ) + I(GroundSF^2)

                      Df Sum of Sq    RSS     Cp
+ factor(BasementHt)   4    160720 565990 455.47
+ I(YearBuilt^2)       1    157397 569313 455.53
+ YearBuilt            1    157382 569329 455.56
+ BasementSF           1    121924 604786 520.24
+ I(BasementSF^2)      1    112159 614551 538.05
+ factor(HouseStyle)   6    113570 613140 545.48
+ factor(BasementFin)  6     92363 634347 584.16
+ I(GarageCars^2)      1     66215 660495 621.86
+ GarageCars           1     63555 663155 626.71
+ factor(KitchenQ)     3     56948 669762 642.77
+ factor(Condition)    7     53872 672838 656.38
+ WoodDeckSF           1     33869 692841 680.86
+ LotArea              1     30692 696018 686.66
+ I(WoodDeckSF^2)      1     16874 709836 711.87
+ TotalRooms           1     13037 713673 718.87
+ I(LotArea^2)         1     11312 715398 722.01
+ FullBath             1      6776 719934 730.29
+ GroundSF             1      3391 723319 736.46
<none>                             726710 740.65

Step:  AIC=455.47
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt)

                      Df Sum of Sq    RSS     Cp
+ factor(HouseStyle)   6     84078 481912 314.09
+ BasementSF           1     74839 491150 320.95
+ YearBuilt            1     66808 499182 335.60
+ I(YearBuilt^2)       1     66606 499384 335.96
+ I(BasementSF^2)      1     65947 500043 337.17
+ GarageCars           1     37702 528288 388.69
+ I(GarageCars^2)      1     37418 528572 389.21
+ factor(BasementFin)  5     39757 526233 392.94
+ LotArea              1     34700 531290 394.17
+ factor(Condition)    7     37467 528522 401.12
+ factor(KitchenQ)     3     29273 536717 408.07
+ WoodDeckSF           1     14324 551666 431.34
+ I(LotArea^2)         1     13501 552489 432.84
+ I(WoodDeckSF^2)      1     10439 555551 438.42
+ TotalRooms           1      5224 560766 447.94
+ GroundSF             1      3983 562007 450.20
<none>                             565990 455.47
+ FullBath             1       200 565790 457.10

Step:  AIC=314.09
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle)

                      Df Sum of Sq    RSS     Cp
+ factor(Condition)    7     38427 443485 257.99
+ YearBuilt            1     30128 451783 261.13
+ I(YearBuilt^2)       1     30095 451816 261.19
+ factor(BasementFin)  5     26815 455097 275.18
+ factor(KitchenQ)     3     24506 457406 275.39
+ GarageCars           1     20826 461085 278.10
+ I(GarageCars^2)      1     20337 461574 278.99
+ BasementSF           1     19221 462690 281.03
+ I(BasementSF^2)      1     13778 468134 290.96
+ GroundSF             1     12741 469171 292.85
+ LotArea              1     12573 469338 293.16
+ WoodDeckSF           1      8474 473438 300.63
+ I(WoodDeckSF^2)      1      4728 477184 307.47
+ I(LotArea^2)         1      2753 479159 311.07
<none>                             481912 314.09
+ TotalRooms           1      1025 480887 314.22
+ FullBath             1       125 481787 315.86

Step:  AIC=257.99
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition)

                      Df Sum of Sq    RSS     Cp
+ I(YearBuilt^2)       1     47529 395956 173.29
+ YearBuilt            1     47420 396065 173.49
+ BasementSF           1     25980 417505 212.60
+ I(GarageCars^2)      1     21214 422271 221.30
+ GarageCars           1     21029 422456 221.63
+ I(BasementSF^2)      1     19924 423561 223.65
+ factor(BasementFin)  5     20154 423331 231.23
+ factor(KitchenQ)     3     16659 426826 233.61
+ LotArea              1     14369 429116 233.78
+ GroundSF             1     13497 429988 235.37
+ WoodDeckSF           1      5209 438276 250.49
+ I(LotArea^2)         1      3787 439697 253.09
+ I(WoodDeckSF^2)      1      2155 441329 256.06
<none>                             443485 257.99
+ TotalRooms           1       437 443048 259.20
+ FullBath             1        97 443388 259.82

Step:  AIC=173.29
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2)

                      Df Sum of Sq    RSS     Cp
+ BasementSF           1   19141.4 376814 140.38
+ LotArea              1   17162.5 378793 143.99
+ I(BasementSF^2)      1   15212.6 380743 147.54
+ I(GarageCars^2)      1   14269.5 381686 149.26
+ GarageCars           1   11136.3 384819 154.98
+ GroundSF             1   10966.0 384990 155.29
+ factor(KitchenQ)     3   11718.3 384237 157.92
+ factor(BasementFin)  5   13148.7 382807 159.31
+ I(LotArea^2)         1    6413.0 389543 163.59
+ WoodDeckSF           1    5133.3 390822 165.93
+ I(WoodDeckSF^2)      1    2603.8 393352 170.54
+ FullBath             1    1859.1 394097 171.90
<none>                             395956 173.29
+ TotalRooms           1     648.3 395307 174.11
+ YearBuilt            1     402.5 395553 174.56

Step:  AIC=140.38
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF

                      Df Sum of Sq    RSS     Cp
+ LotArea              1   15952.0 360862 113.28
+ I(GarageCars^2)      1    9569.7 367245 124.92
+ factor(KitchenQ)     3   11622.0 365192 125.18
+ GarageCars           1    7546.2 369268 128.61
+ factor(BasementFin)  5   11477.0 365337 129.44
+ I(LotArea^2)         1    6477.0 370337 130.56
+ WoodDeckSF           1    3869.5 372945 135.32
+ FullBath             1    3450.1 373364 136.08
+ GroundSF             1    3147.1 373667 136.63
+ I(WoodDeckSF^2)      1    1520.3 375294 139.60
+ TotalRooms           1    1403.4 375411 139.82
<none>                             376814 140.38
+ I(BasementSF^2)      1     578.7 376236 141.32
+ YearBuilt            1     324.7 376490 141.78

Step:  AIC=113.28
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea

                      Df Sum of Sq    RSS      Cp
+ I(LotArea^2)         1   10707.5 350155  95.744
+ factor(KitchenQ)     3   12062.7 348800  97.272
+ I(GarageCars^2)      1    7903.0 352959 100.860
+ factor(BasementFin)  5   11062.9 349799 103.096
+ GarageCars           1    5588.2 355274 105.083
+ FullBath             1    3607.4 357255 108.696
+ WoodDeckSF           1    3145.9 357716 109.538
+ TotalRooms           1    2410.8 358452 110.879
<none>                             360862 113.277
+ GroundSF             1     902.6 359960 113.630
+ I(WoodDeckSF^2)      1     870.8 359991 113.688
+ YearBuilt            1     592.9 360269 114.195
+ I(BasementSF^2)      1     353.9 360508 114.631

Step:  AIC=95.74
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2)

                      Df Sum of Sq    RSS     Cp
+ factor(KitchenQ)     3   11542.5 338612 80.689
+ factor(BasementFin)  5   12613.1 337542 82.736
+ I(GarageCars^2)      1    7747.1 342408 83.612
+ GarageCars           1    5656.0 344499 87.427
+ TotalRooms           1    4897.2 345258 88.811
+ FullBath             1    4220.3 345935 90.046
+ WoodDeckSF           1    2703.5 347451 92.812
<none>                             350155 95.744
+ I(WoodDeckSF^2)      1     896.4 349258 96.109
+ GroundSF             1     696.8 349458 96.473
+ YearBuilt            1     373.0 349782 97.064
+ I(BasementSF^2)      1       3.7 350151 97.737

Step:  AIC=80.69
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ)

                      Df Sum of Sq    RSS     Cp
+ I(GarageCars^2)      1    7655.1 330957 68.724
+ factor(BasementFin)  5   11288.4 327324 70.097
+ GarageCars           1    5847.9 332764 72.021
+ TotalRooms           1    4509.9 334102 74.462
+ FullBath             1    4213.5 334399 75.002
+ WoodDeckSF           1    2158.8 336454 78.751
<none>                             338612 80.689
+ GroundSF             1     832.9 337779 81.169
+ I(WoodDeckSF^2)      1     822.9 337789 81.188
+ YearBuilt            1     106.8 338505 82.494
+ I(BasementSF^2)      1       3.5 338609 82.682

Step:  AIC=68.72
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2)

                      Df Sum of Sq    RSS     Cp
+ factor(BasementFin)  5   13552.3 317405 54.002
+ TotalRooms           1    5177.0 325780 61.281
+ FullBath             1    4754.9 326202 62.051
+ WoodDeckSF           1    1838.1 329119 67.371
<none>                             330957 68.724
+ GroundSF             1     918.9 330038 69.048
+ I(WoodDeckSF^2)      1     730.4 330227 69.392
+ GarageCars           1     139.3 330818 70.470
+ I(BasementSF^2)      1      42.2 330915 70.647
+ YearBuilt            1       7.4 330950 70.711

Step:  AIC=54
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin)

                  Df Sum of Sq    RSS     Cp
+ FullBath         1   3012.04 314393 50.508
+ TotalRooms       1   2505.48 314899 51.432
+ GroundSF         1   1836.90 315568 52.652
+ WoodDeckSF       1   1111.30 316293 53.975
<none>                         317405 54.002
+ YearBuilt        1    485.74 316919 55.116
+ GarageCars       1    442.96 316962 55.194
+ I(WoodDeckSF^2)  1    296.71 317108 55.461
+ I(BasementSF^2)  1    111.45 317293 55.799

Step:  AIC=50.51
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath

                  Df Sum of Sq    RSS     Cp
+ GroundSF         1    4001.2 310392 45.209
+ TotalRooms       1    1470.8 312922 49.825
<none>                         314393 50.508
+ WoodDeckSF       1     991.6 313401 50.699
+ YearBuilt        1     939.9 313453 50.793
+ GarageCars       1     365.1 314028 51.842
+ I(WoodDeckSF^2)  1     220.8 314172 52.105
+ I(BasementSF^2)  1     126.7 314266 52.277

Step:  AIC=45.21
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF

                  Df Sum of Sq    RSS     Cp
+ TotalRooms       1    3565.6 306826 40.705
+ WoodDeckSF       1    1128.4 309263 45.151
<none>                         310392 45.209
+ GarageCars       1     987.8 309404 45.407
+ YearBuilt        1     906.8 309485 45.555
+ I(WoodDeckSF^2)  1     311.9 310080 46.640
+ I(BasementSF^2)  1      78.1 310313 47.067

Step:  AIC=40.7
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms

                  Df Sum of Sq    RSS     Cp
+ GarageCars       1   1346.69 305479 40.248
+ YearBuilt        1   1184.41 305642 40.544
+ WoodDeckSF       1   1114.54 305711 40.672
<none>                         306826 40.705
+ I(WoodDeckSF^2)  1    281.70 306544 42.191
+ I(BasementSF^2)  1     61.43 306765 42.593

Step:  AIC=40.25
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars

                  Df Sum of Sq    RSS     Cp
+ WoodDeckSF       1   1113.64 304366 40.217
<none>                         305479 40.248
+ YearBuilt        1    857.57 304622 40.684
+ I(WoodDeckSF^2)  1    290.63 305189 41.718
+ I(BasementSF^2)  1     48.76 305431 42.159

Step:  AIC=40.22
Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars + WoodDeckSF

                  Df Sum of Sq    RSS     Cp
<none>                         304366 40.217
+ I(WoodDeckSF^2)  1    935.01 303431 40.511
+ YearBuilt        1    887.06 303479 40.599
+ I(BasementSF^2)  1     17.03 304349 42.186

Call:
lm(formula = Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars + WoodDeckSF, data = AmesTrain6a)

Coefficients:
             (Intercept)       factor(ExteriorQ)Fa       factor(ExteriorQ)Gd       factor(ExteriorQ)TA  
              -3.578e+02                -7.652e+01                -4.455e+01                -6.249e+01  
           I(GroundSF^2)      factor(BasementHt)Fa      factor(BasementHt)Gd    factor(BasementHt)None  
               8.318e-06                -4.839e+01                -3.652e+01                -5.351e+01  
    factor(BasementHt)TA  factor(HouseStyle)1.5Unf  factor(HouseStyle)1Story  factor(HouseStyle)2.5Unf  
              -4.732e+01                 2.897e+01                 1.281e+01                 1.156e+01  
factor(HouseStyle)2Story  factor(HouseStyle)SFoyer    factor(HouseStyle)SLvl        factor(Condition)3  
               4.640e+00                 1.714e+00                 8.855e+00                 5.008e+00  
      factor(Condition)4        factor(Condition)5        factor(Condition)6        factor(Condition)7  
               1.510e+01                 2.705e+01                 3.544e+01                 4.263e+01  
      factor(Condition)8        factor(Condition)9            I(YearBuilt^2)                BasementSF  
               5.135e+01                 6.339e+01                 1.281e-04                 1.360e-02  
                 LotArea              I(LotArea^2)        factor(KitchenQ)Fa        factor(KitchenQ)Gd  
               4.725e-03                -1.464e-07                -3.115e+01                -2.127e+01  
      factor(KitchenQ)TA           I(GarageCars^2)    factor(BasementFin)BLQ    factor(BasementFin)GLQ  
              -2.217e+01                 4.559e+00                 7.728e-01                 3.401e+00  
  factor(BasementFin)LwQ   factor(BasementFin)None    factor(BasementFin)Rec    factor(BasementFin)Unf  
              -9.447e+00                        NA                -9.769e-01                -7.594e+00  
                FullBath                  GroundSF                TotalRooms                GarageCars  
              -7.362e+00                 5.274e-02                -3.331e+00                -7.686e+00  
              WoodDeckSF  
               1.299e-02  

Backward selection

MSE=(summary(modTransformFull)$sigma)^2
step(modTransformFull,scale=MSE)
Start:  AIC=43
Price ~ LotArea + I(LotArea^2) + YearBuilt + I(YearBuilt^2) + 
    BasementSF + I(BasementSF^2) + GarageCars + I(GarageCars^2) + 
    WoodDeckSF + I(WoodDeckSF^2) + GroundSF + I(GroundSF^2) + 
    FullBath + TotalRooms + factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(KitchenQ) + factor(BasementHt) + 
    factor(Condition)

                      Df Sum of Sq    RSS      Cp
- I(BasementSF^2)      1        63 302665  41.115
- BasementSF           1       198 302800  41.361
- YearBuilt            1       759 303361  42.384
- I(WoodDeckSF^2)      1       861 303463  42.570
- I(YearBuilt^2)       1       872 303474  42.590
- GarageCars           1       990 303592  42.805
<none>                             302602  43.000
- WoodDeckSF           1      1663 304265  44.033
- factor(HouseStyle)   6      7307 309909  44.329
- I(GroundSF^2)        1      3281 305884  46.986
- TotalRooms           1      4175 306777  48.616
- I(GarageCars^2)      1      4235 306837  48.725
- FullBath             1      4376 306979  48.983
- factor(KitchenQ)     3      8634 311236  52.750
- factor(BasementFin)  5     11177 313779  53.389
- GroundSF             1      6894 309496  53.576
- I(LotArea^2)         1     12250 314853  63.347
- LotArea              1     19813 322415  77.142
- factor(ExteriorQ)    3     27985 330587  88.049
- factor(BasementHt)   3     28051 330653  88.170
- factor(Condition)    7     43156 345758 107.724

Step:  AIC=41.12
Price ~ LotArea + I(LotArea^2) + YearBuilt + I(YearBuilt^2) + 
    BasementSF + GarageCars + I(GarageCars^2) + WoodDeckSF + 
    I(WoodDeckSF^2) + GroundSF + I(GroundSF^2) + FullBath + TotalRooms + 
    factor(HouseStyle) + factor(ExteriorQ) + factor(BasementFin) + 
    factor(KitchenQ) + factor(BasementHt) + factor(Condition)

                      Df Sum of Sq    RSS      Cp
- YearBuilt            1       765 303431  40.511
- I(WoodDeckSF^2)      1       813 303479  40.599
- I(YearBuilt^2)       1       879 303544  40.718
- GarageCars           1      1002 303668  40.943
<none>                             302665  41.115
- WoodDeckSF           1      1619 304285  42.069
- factor(HouseStyle)   6      7268 309934  42.374
- I(GroundSF^2)        1      3248 305914  45.041
- BasementSF           1      3892 306558  46.215
- TotalRooms           1      4192 306858  46.763
- I(GarageCars^2)      1      4276 306941  46.915
- FullBath             1      4372 307038  47.091
- factor(KitchenQ)     3      8646 311311  50.887
- factor(BasementFin)  5     11159 313824  51.471
- GroundSF             1      6982 309647  51.851
- I(LotArea^2)         1     12285 314951  61.526
- LotArea              1     20025 322690  75.644
- factor(ExteriorQ)    3     28333 330998  86.800
- factor(BasementHt)   3     29077 331742  88.157
- factor(Condition)    7     43098 345764 105.734

Step:  AIC=40.51
Price ~ LotArea + I(LotArea^2) + I(YearBuilt^2) + BasementSF + 
    GarageCars + I(GarageCars^2) + WoodDeckSF + I(WoodDeckSF^2) + 
    GroundSF + I(GroundSF^2) + FullBath + TotalRooms + factor(HouseStyle) + 
    factor(ExteriorQ) + factor(BasementFin) + factor(KitchenQ) + 
    factor(BasementHt) + factor(Condition)

                      Df Sum of Sq    RSS      Cp
- I(WoodDeckSF^2)      1       935 304366  40.217
<none>                             303431  40.511
- GarageCars           1      1306 304737  40.894
- factor(HouseStyle)   6      7176 310607  41.602
- WoodDeckSF           1      1758 305189  41.718
- I(GroundSF^2)        1      3073 306504  44.117
- BasementSF           1      3862 307293  45.556
- FullBath             1      3980 307411  45.772
- TotalRooms           1      4005 307436  45.817
- I(GarageCars^2)      1      5110 308541  47.833
- factor(BasementFin)  5     10521 313952  49.704
- factor(KitchenQ)     3      8931 312361  50.802
- GroundSF             1      7056 310487  51.383
- I(LotArea^2)         1     12375 315806  61.086
- LotArea              1     19968 323399  74.937
- factor(BasementHt)   3     31977 335407  92.842
- I(YearBuilt^2)       1     29816 333247  92.901
- factor(ExteriorQ)    3     32715 336146  94.190
- factor(Condition)    7     42338 345769 103.743

Step:  AIC=40.22
Price ~ LotArea + I(LotArea^2) + I(YearBuilt^2) + BasementSF + 
    GarageCars + I(GarageCars^2) + WoodDeckSF + GroundSF + I(GroundSF^2) + 
    FullBath + TotalRooms + factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(KitchenQ) + factor(BasementHt) + 
    factor(Condition)

                      Df Sum of Sq    RSS      Cp
<none>                             304366  40.217
- WoodDeckSF           1      1114 305479  40.248
- GarageCars           1      1346 305711  40.672
- factor(HouseStyle)   6      7179 311544  41.312
- I(GroundSF^2)        1      2870 307235  43.452
- BasementSF           1      3580 307946  44.747
- TotalRooms           1      3910 308275  45.349
- FullBath             1      3929 308295  45.384
- I(GarageCars^2)      1      5246 309611  47.786
- factor(BasementFin)  5     10450 314816  49.280
- GroundSF             1      7200 311566  51.351
- factor(KitchenQ)     3      9474 313840  51.499
- I(LotArea^2)         1     13021 317387  61.970
- LotArea              1     20431 324797  75.487
- I(YearBuilt^2)       1     30239 334604  93.377
- factor(ExteriorQ)    3     33011 337376  94.434
- factor(BasementHt)   3     34043 338408  96.317
- factor(Condition)    7     41713 346079 102.309

Call:
lm(formula = Price ~ LotArea + I(LotArea^2) + I(YearBuilt^2) + 
    BasementSF + GarageCars + I(GarageCars^2) + WoodDeckSF + 
    GroundSF + I(GroundSF^2) + FullBath + TotalRooms + factor(HouseStyle) + 
    factor(ExteriorQ) + factor(BasementFin) + factor(KitchenQ) + 
    factor(BasementHt) + factor(Condition), data = AmesTrain6a)

Coefficients:
             (Intercept)                   LotArea              I(LotArea^2)            I(YearBuilt^2)  
              -3.578e+02                 4.725e-03                -1.464e-07                 1.281e-04  
              BasementSF                GarageCars           I(GarageCars^2)                WoodDeckSF  
               1.360e-02                -7.686e+00                 4.559e+00                 1.299e-02  
                GroundSF             I(GroundSF^2)                  FullBath                TotalRooms  
               5.274e-02                 8.318e-06                -7.362e+00                -3.331e+00  
factor(HouseStyle)1.5Unf  factor(HouseStyle)1Story  factor(HouseStyle)2.5Unf  factor(HouseStyle)2Story  
               2.897e+01                 1.281e+01                 1.156e+01                 4.640e+00  
factor(HouseStyle)SFoyer    factor(HouseStyle)SLvl       factor(ExteriorQ)Fa       factor(ExteriorQ)Gd  
               1.714e+00                 8.855e+00                -7.652e+01                -4.455e+01  
     factor(ExteriorQ)TA    factor(BasementFin)BLQ    factor(BasementFin)GLQ    factor(BasementFin)LwQ  
              -6.249e+01                 7.728e-01                 3.401e+00                -9.447e+00  
 factor(BasementFin)None    factor(BasementFin)Rec    factor(BasementFin)Unf        factor(KitchenQ)Fa  
              -5.351e+01                -9.769e-01                -7.594e+00                -3.115e+01  
      factor(KitchenQ)Gd        factor(KitchenQ)TA      factor(BasementHt)Fa      factor(BasementHt)Gd  
              -2.127e+01                -2.217e+01                -4.839e+01                -3.652e+01  
  factor(BasementHt)None      factor(BasementHt)TA        factor(Condition)3        factor(Condition)4  
                      NA                -4.732e+01                 5.008e+00                 1.510e+01  
      factor(Condition)5        factor(Condition)6        factor(Condition)7        factor(Condition)8  
               2.705e+01                 3.544e+01                 4.263e+01                 5.135e+01  
      factor(Condition)9  
               6.339e+01  

Backward selection

BackwardMod = lm(Price ~ LotArea + I(LotArea^2) + I(YearBuilt^2) + BasementSF + 
    GarageCars + I(GarageCars^2) + WoodDeckSF + GroundSF + I(GroundSF^2) + 
    FullBath + TotalRooms + factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(KitchenQ) + factor(BasementHt) + 
    factor(Condition), data=AmesTrain6a)
summary(BackwardMod)

Call:
lm(formula = Price ~ LotArea + I(LotArea^2) + I(YearBuilt^2) + 
    BasementSF + GarageCars + I(GarageCars^2) + WoodDeckSF + 
    GroundSF + I(GroundSF^2) + FullBath + TotalRooms + factor(HouseStyle) + 
    factor(ExteriorQ) + factor(BasementFin) + factor(KitchenQ) + 
    factor(BasementHt) + factor(Condition), data = AmesTrain6a)

Residuals:
    Min      1Q  Median      3Q     Max 
-84.323 -11.695  -1.128  11.311 111.963 

Coefficients: (1 not defined because of singularities)
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)              -3.578e+02  7.287e+01  -4.910 1.20e-06 ***
LotArea                   4.725e-03  7.742e-04   6.104 1.95e-09 ***
I(LotArea^2)             -1.464e-07  3.004e-08  -4.873 1.44e-06 ***
I(YearBuilt^2)            1.281e-04  1.726e-05   7.426 4.26e-13 ***
BasementSF                1.360e-02  5.324e-03   2.555 0.010885 *  
GarageCars               -7.686e+00  4.907e+00  -1.567 0.117796    
I(GarageCars^2)           4.559e+00  1.474e+00   3.093 0.002083 ** 
WoodDeckSF                1.299e-02  9.113e-03   1.425 0.154713    
GroundSF                  5.274e-02  1.456e-02   3.623 0.000318 ***
I(GroundSF^2)             8.318e-06  3.636e-06   2.287 0.022543 *  
FullBath                 -7.362e+00  2.750e+00  -2.677 0.007657 ** 
TotalRooms               -3.331e+00  1.248e+00  -2.670 0.007805 ** 
factor(HouseStyle)1.5Unf  2.897e+01  1.160e+01   2.497 0.012826 *  
factor(HouseStyle)1Story  1.281e+01  4.702e+00   2.725 0.006637 ** 
factor(HouseStyle)2.5Unf  1.156e+01  1.461e+01   0.791 0.429161    
factor(HouseStyle)2Story  4.640e+00  4.357e+00   1.065 0.287368    
factor(HouseStyle)SFoyer  1.714e+00  7.480e+00   0.229 0.818817    
factor(HouseStyle)SLvl    8.855e+00  5.570e+00   1.590 0.112434    
factor(ExteriorQ)Fa      -7.652e+01  1.346e+01  -5.684 2.13e-08 ***
factor(ExteriorQ)Gd      -4.455e+01  8.171e+00  -5.452 7.52e-08 ***
factor(ExteriorQ)TA      -6.249e+01  8.774e+00  -7.122 3.30e-12 ***
factor(BasementFin)BLQ    7.728e-01  4.204e+00   0.184 0.854217    
factor(BasementFin)GLQ    3.401e+00  3.363e+00   1.011 0.312315    
factor(BasementFin)LwQ   -9.447e+00  5.284e+00  -1.788 0.074359 .  
factor(BasementFin)None  -5.351e+01  1.088e+01  -4.918 1.15e-06 ***
factor(BasementFin)Rec   -9.769e-01  4.501e+00  -0.217 0.828267    
factor(BasementFin)Unf   -7.594e+00  3.189e+00  -2.382 0.017571 *  
factor(KitchenQ)Fa       -3.115e+01  8.731e+00  -3.568 0.000391 ***
factor(KitchenQ)Gd       -2.127e+01  5.698e+00  -3.733 0.000209 ***
factor(KitchenQ)TA       -2.217e+01  5.868e+00  -3.779 0.000175 ***
factor(BasementHt)Fa     -4.839e+01  8.853e+00  -5.465 6.99e-08 ***
factor(BasementHt)Gd     -3.652e+01  5.092e+00  -7.172 2.37e-12 ***
factor(BasementHt)None           NA         NA      NA       NA    
factor(BasementHt)TA     -4.732e+01  6.135e+00  -7.714 5.70e-14 ***
factor(Condition)3        5.008e+00  2.711e+01   0.185 0.853526    
factor(Condition)4        1.510e+01  2.651e+01   0.570 0.569197    
factor(Condition)5        2.705e+01  2.626e+01   1.030 0.303407    
factor(Condition)6        3.544e+01  2.632e+01   1.346 0.178711    
factor(Condition)7        4.263e+01  2.634e+01   1.618 0.106134    
factor(Condition)8        5.135e+01  2.641e+01   1.945 0.052332 .  
factor(Condition)9        6.339e+01  2.722e+01   2.329 0.020235 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 23.42 on 555 degrees of freedom
Multiple R-squared:  0.8948,    Adjusted R-squared:  0.8874 
F-statistic: 121.1 on 39 and 555 DF,  p-value: < 2.2e-16
StepwiseMod = lm(Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars + WoodDeckSF, data=AmesTrain6a)
summary(StepwiseMod)

Call:
lm(formula = Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars + WoodDeckSF, data = AmesTrain6a)

Residuals:
    Min      1Q  Median      3Q     Max 
-84.323 -11.695  -1.128  11.311 111.963 

Coefficients: (1 not defined because of singularities)
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)              -3.578e+02  7.287e+01  -4.910 1.20e-06 ***
factor(ExteriorQ)Fa      -7.652e+01  1.346e+01  -5.684 2.13e-08 ***
factor(ExteriorQ)Gd      -4.455e+01  8.171e+00  -5.452 7.52e-08 ***
factor(ExteriorQ)TA      -6.249e+01  8.774e+00  -7.122 3.30e-12 ***
I(GroundSF^2)             8.318e-06  3.636e-06   2.287 0.022543 *  
factor(BasementHt)Fa     -4.839e+01  8.853e+00  -5.465 6.99e-08 ***
factor(BasementHt)Gd     -3.652e+01  5.092e+00  -7.172 2.37e-12 ***
factor(BasementHt)None   -5.351e+01  1.088e+01  -4.918 1.15e-06 ***
factor(BasementHt)TA     -4.732e+01  6.135e+00  -7.714 5.70e-14 ***
factor(HouseStyle)1.5Unf  2.897e+01  1.160e+01   2.497 0.012826 *  
factor(HouseStyle)1Story  1.281e+01  4.702e+00   2.725 0.006637 ** 
factor(HouseStyle)2.5Unf  1.156e+01  1.461e+01   0.791 0.429161    
factor(HouseStyle)2Story  4.640e+00  4.357e+00   1.065 0.287368    
factor(HouseStyle)SFoyer  1.714e+00  7.480e+00   0.229 0.818817    
factor(HouseStyle)SLvl    8.855e+00  5.570e+00   1.590 0.112434    
factor(Condition)3        5.008e+00  2.711e+01   0.185 0.853526    
factor(Condition)4        1.510e+01  2.651e+01   0.570 0.569197    
factor(Condition)5        2.705e+01  2.626e+01   1.030 0.303407    
factor(Condition)6        3.544e+01  2.632e+01   1.346 0.178711    
factor(Condition)7        4.263e+01  2.634e+01   1.618 0.106134    
factor(Condition)8        5.135e+01  2.641e+01   1.945 0.052332 .  
factor(Condition)9        6.339e+01  2.722e+01   2.329 0.020235 *  
I(YearBuilt^2)            1.281e-04  1.726e-05   7.426 4.26e-13 ***
BasementSF                1.360e-02  5.324e-03   2.555 0.010885 *  
LotArea                   4.725e-03  7.742e-04   6.104 1.95e-09 ***
I(LotArea^2)             -1.464e-07  3.004e-08  -4.873 1.44e-06 ***
factor(KitchenQ)Fa       -3.115e+01  8.731e+00  -3.568 0.000391 ***
factor(KitchenQ)Gd       -2.127e+01  5.698e+00  -3.733 0.000209 ***
factor(KitchenQ)TA       -2.217e+01  5.868e+00  -3.779 0.000175 ***
I(GarageCars^2)           4.559e+00  1.474e+00   3.093 0.002083 ** 
factor(BasementFin)BLQ    7.728e-01  4.204e+00   0.184 0.854217    
factor(BasementFin)GLQ    3.401e+00  3.363e+00   1.011 0.312315    
factor(BasementFin)LwQ   -9.447e+00  5.284e+00  -1.788 0.074359 .  
factor(BasementFin)None          NA         NA      NA       NA    
factor(BasementFin)Rec   -9.769e-01  4.501e+00  -0.217 0.828267    
factor(BasementFin)Unf   -7.594e+00  3.189e+00  -2.382 0.017571 *  
FullBath                 -7.362e+00  2.750e+00  -2.677 0.007657 ** 
GroundSF                  5.274e-02  1.456e-02   3.623 0.000318 ***
TotalRooms               -3.331e+00  1.248e+00  -2.670 0.007805 ** 
GarageCars               -7.686e+00  4.907e+00  -1.567 0.117796    
WoodDeckSF                1.299e-02  9.113e-03   1.425 0.154713    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 23.42 on 555 degrees of freedom
Multiple R-squared:  0.8948,    Adjusted R-squared:  0.8874 
F-statistic: 121.1 on 39 and 555 DF,  p-value: < 2.2e-16
ForwardMod= lm(Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars, data=AmesTrain6a)
summary(ForwardMod)

Call:
lm(formula = Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars, data = AmesTrain6a)

Residuals:
    Min      1Q  Median      3Q     Max 
-82.839 -12.255  -0.778  11.658 110.462 

Coefficients: (1 not defined because of singularities)
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)              -3.536e+02  7.288e+01  -4.851 1.59e-06 ***
factor(ExteriorQ)Fa      -7.716e+01  1.347e+01  -5.729 1.66e-08 ***
factor(ExteriorQ)Gd      -4.493e+01  8.174e+00  -5.496 5.92e-08 ***
factor(ExteriorQ)TA      -6.272e+01  8.780e+00  -7.143 2.87e-12 ***
I(GroundSF^2)             8.590e-06  3.635e-06   2.363 0.018454 *  
factor(BasementHt)Fa     -4.930e+01  8.838e+00  -5.578 3.81e-08 ***
factor(BasementHt)Gd     -3.667e+01  5.096e+00  -7.196 2.01e-12 ***
factor(BasementHt)None   -5.439e+01  1.087e+01  -5.003 7.60e-07 ***
factor(BasementHt)TA     -4.812e+01  6.114e+00  -7.871 1.85e-14 ***
factor(HouseStyle)1.5Unf  2.831e+01  1.160e+01   2.439 0.015028 *  
factor(HouseStyle)1Story  1.286e+01  4.706e+00   2.733 0.006474 ** 
factor(HouseStyle)2.5Unf  1.108e+01  1.462e+01   0.758 0.448730    
factor(HouseStyle)2Story  4.800e+00  4.360e+00   1.101 0.271413    
factor(HouseStyle)SFoyer  1.938e+00  7.485e+00   0.259 0.795765    
factor(HouseStyle)SLvl    9.936e+00  5.523e+00   1.799 0.072551 .  
factor(Condition)3        4.885e+00  2.714e+01   0.180 0.857209    
factor(Condition)4        1.515e+01  2.654e+01   0.571 0.568369    
factor(Condition)5        2.728e+01  2.628e+01   1.038 0.299756    
factor(Condition)6        3.565e+01  2.635e+01   1.353 0.176596    
factor(Condition)7        4.295e+01  2.636e+01   1.629 0.103845    
factor(Condition)8        5.208e+01  2.643e+01   1.971 0.049257 *  
factor(Condition)9        6.437e+01  2.724e+01   2.363 0.018448 *  
I(YearBuilt^2)            1.274e-04  1.726e-05   7.378 5.86e-13 ***
BasementSF                1.406e-02  5.319e-03   2.643 0.008446 ** 
LotArea                   4.789e-03  7.736e-04   6.191 1.17e-09 ***
I(LotArea^2)             -1.482e-07  3.004e-08  -4.934 1.07e-06 ***
factor(KitchenQ)Fa       -3.194e+01  8.721e+00  -3.662 0.000274 ***
factor(KitchenQ)Gd       -2.141e+01  5.702e+00  -3.754 0.000192 ***
factor(KitchenQ)TA       -2.246e+01  5.869e+00  -3.827 0.000144 ***
I(GarageCars^2)           4.595e+00  1.475e+00   3.115 0.001935 ** 
factor(BasementFin)BLQ    8.770e-01  4.207e+00   0.208 0.834937    
factor(BasementFin)GLQ    3.297e+00  3.366e+00   0.980 0.327636    
factor(BasementFin)LwQ   -9.818e+00  5.282e+00  -1.859 0.063603 .  
factor(BasementFin)None          NA         NA      NA       NA    
factor(BasementFin)Rec   -1.153e+00  4.504e+00  -0.256 0.798004    
factor(BasementFin)Unf   -7.892e+00  3.185e+00  -2.478 0.013503 *  
FullBath                 -7.445e+00  2.752e+00  -2.705 0.007040 ** 
GroundSF                  5.212e-02  1.456e-02   3.579 0.000375 ***
TotalRooms               -3.337e+00  1.249e+00  -2.673 0.007747 ** 
GarageCars               -7.689e+00  4.911e+00  -1.566 0.118012    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 23.44 on 556 degrees of freedom
Multiple R-squared:  0.8944,    Adjusted R-squared:  0.8872 
F-statistic:   124 on 38 and 556 DF,  p-value: < 2.2e-16

The forward selection model has the best parsimony and suggests a 16-variable mod with an r-squared value of .887, an AIC of 40.22, and a Mallow CP of ~40. (This is one fewer variable but a slightly worse (.002) adjusted r-squared than the models suggested by stepwise and backward selection.) We chose the backward selection model because it uses the fewest variables, which is best for parsimony, especially given how many variables we could use.

Part 8: Cross-Validation

Redo the cross-validation analysis with your test data for your new fancy model.

mean(ForwardMod$residuals)
[1] -1.845529e-16
sd(ForwardMod$residuals)
[1] 22.53261
mean(modTransformFull$residuals)
[1] 4.097209e-16
sd(modTransformFull$residuals)
[1] 22.57059
plot(ForwardMod$residuals)
abline(0,0)

ShrunkenMod=lm(Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars, data=AmesTrain6)
RefitAmes=predict.lm(ShrunkenMod, newdata=AmesTest6)
prediction from a rank-deficient fit may be misleading
cor(AmesTest6$Price,RefitAmes)
[1] 0.9135647
crosscorr=cor(AmesTest6$Price,RefitAmes)
cor(log(AmesTest6$Price),RefitAmes)
[1] 0.9045966
crosscorr=cor(AmesTest6$Price,RefitAmes)
crosscorr^2
[1] 0.8346004
.8872-crosscorr^2
[1] 0.05259962

Discuss mean of residuals, std. dev of residuals, cross-validation correlation, and shrinkage

The chunk beginning at line 312 shows us that having a model with fewer variables (ForwardMod) does not dramatically impact the residual values of either the mean or the standard deviation. The mean residuals of the ForwardMod is very close to zero, indicating that there is normality in the data. The plot of the residuals backs up this assumption, since it doesn’t show any fanning pattern or skew. The standard deviation is also small relative to the size of the data. These are all good signs and encourage us to use ForwardMod. ForwardMod is a also a more efficent model than the full mod because it has fewer variables for essentially the same r-squared value (.002 difference). The cross-validation of the model also shows shrinkage of the ForwardMod as a healthy sign, since it shows only a .05 difference between the model we made and the holdout data. This means we did not overfit the model.

Part 9: Final Model

Final Changes

We chose not to make any more adjustments to our model, because we think it does a good job balancing the number of variables it uses and predictive ability. This conclusion was supported by normality in the residuals and negative shrinkage. The adjusted r-squared is high and we have relatively good parsimony.

newpredictiondata= data.frame(ExteriorQ="Gd", BasementHt="Ex", Condition=5, YearBuilt=1995, BasementSF=1150, KitchenQ="TA", GarageCars=2, BasementFin="Unf", TotalRooms=9, GroundSF=2314, FullBath=2, HouseStyle="2Story", WoodDeckSF=274, LotArea=11060)
predict.lm(ForwardMod, newpredictiondata, interval="prediction", level=.95)
prediction from a rank-deficient fit may be misleading
      fit     lwr      upr
1 234.182 178.715 289.6491

With a 2 story house from Ames, Iowa, with a good exterior quality, excellent basement height, average overall condition, built in 1995, basement square footage of 1150 ft, average kitchen quality, space for 2 cars in the garage, unfinished basement, 9 total rooms, 2314 ft in living area square feet, 2 full baths, 274 sq ft of wood deck, and 11060 sq ft lot area, we expect the price to be $234,182. We are 95% confident that the price will fall between $178,715 and $289,649.

---
title: "R Notebook"
output: html_notebook
---

### Part 6: Cross Validation
# Compute Predicted Price for Model from parts 1/2

```{r}
firstmodtestdata = lm(Price~LotArea+YearBuilt+BasementSF+FullBath+TotalRooms+GarageCars+WoodDeckSF, data=AmesTest6)
summary(firstmodtestdata)
predict(firstmodtestdata, data.frame(AmesTest6), level = .95, interval = "predict")
```

# Compute the residuals for the 200 holdout cases
```{r}
rstandard(firstmodtestdata)
rstudent(firstmodtestdata)
```
# Compute the mean and standard deviation of these residuals. Are they close to what you expect from the training model?
```{r}
plot(firstmodtestdata$residuals~firstmodtestdata$fitted.values)
abline(a=0, b=0)

mean(firstmodtestdata$residuals)
sd(firstmodtestdata$residuals)

firstmodtraindata = lm(Price~LotArea+YearBuilt+BasementSF+FullBath+TotalRooms+GarageCars+WoodDeckSF, data=AmesTrain6a)
mean(firstmodtraindata$residuals)
sd(firstmodtraindata$residuals)
```
The mean residuals of our simplest model using the test data are very similar to the mean residuals we had with the training data. This is a good sign that our model was not overfitted. Additionally, the standard deviations were similar, which was expected. 

# Are any holdout cases especially poorly predicted by the training model? If so, identify by the row number(s) in the holdout data. 
```{r}
which.max(rstandard(firstmodtestdata))
max(rstandard(firstmodtestdata))
which.max(rstudent(firstmodtestdata))
max(rstudent(firstmodtestdata))
which.min(rstandard(firstmodtestdata))
min(rstandard(firstmodtestdata))
which.min(rstudent(firstmodtestdata))
min(rstudent(firstmodtestdata))
```
Although some of these outliers are over the threshold for concern for rstudent and rstandard residuals, they are not substantially different from the outliers we had in our first model. This means that overfitting is likely not the cause of the outliers of concern. Finally, the outliers don't change substantially between rstandard and rstudent, meaning that their removal from the data set doesn't significantly alter our model. 

# Compute the correlation between the predicted values above and actual prices for the holdout sample. 
```{r}
summary(firstmodtraindata)
fitAmes=predict(firstmodtraindata, newdata=AmesTest6)
holoutresid=(AmesTest6$Price)-fitAmes
mean(holdoutresid)
cor(AmesTest6$Price, fitAmes)
crosscorr=cor(AmesTest6$Price, fitAmes)
crosscorr^2
0.6795-crosscorr^2
```
Our shrinkage is 0.05727804, which indicates that our model fits our test data almost as well as it fitr our training data. This means that we did not overfit our model (which makes sense, because this was the most basic model that we used). 


### Part 7: A Fancy Model
# Categorical variables from the original dataset
```{r}
modTransformCat=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCat)
MSE=(summary(modTransformCat)$sigma)^2
step(none,scope=list(upper=modTransformCat),scale=MSE)
```
```{r}
modCatTransformForward=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
MSE=(summary(modCatTransformForward)$sigma)^2
none=lm(Price~1,data=AmesTrain6a)
step(none,scope=list(upper=modCatTransformForward),scale=MSE, direction = "forward")
```

```{r}
modTransformCatBackward=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
MSE=(summary(modTransformCatBackward)$sigma)^2
step(modTransformCatBackward,scale=MSE)
```
```{r}
modCatReduced = lm (Price ~ factor(ExteriorQ) + factor(BasementHt) + 
    factor(GarageType) + factor(KitchenQ) + factor(HouseStyle) + 
    factor(LotConfig) + factor(Condition), data = AmesTrain6a)

modCatFull=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)

Cp455(modCatReduced, modCatFull)
```
After doing forward, backward, and stepwise regression, we're going to use the categorical models that forward and stepwise gave us because of the low AIC and Mallow Cp values. The AIC for forward and stepwise was 36.37 and the AIC for backward selection was 36.25. Although the backward selection AIC was slightly lower, we chose to use the factors suggested by forward and stepwise selection because they agreed, and also because we may do a second round of selection when we combine these variables with the numerical variables. This means that we will include  ExteriorQ, BasementHt, GarageType, KitchenQ, HouseStyle, LotConfig, and Condition in the next model. Additionally, the CP for the model suggested by stepwise selection is 36.369, which is substantially more than the number of variables in the model, which means the model isn't very efficient. With that said, this CP is about the same between forward, backward, and stepwise selection, and may improve when we include numerical variables. Furthermore, a higher CP makes sense for when this many variables are introduced into a model. 

# Transformations of predictors.
We chose these predictors to transform based on the stepwise, backward, and forward selection that we did for Assignment #3
```{r}
modLotArea=lm(Price~LotArea, data=AmesTrain6a)
modLotAreaSquared=lm(Price~LotArea+I(LotArea^2), data=AmesTrain6a)
modLotAreaSqrt=lm(Price~LotArea+I(sqrt(LotArea)), data=AmesTrain6a)
modLotAreaLog=lm(Price~(log(LotArea)), data=AmesTrain6a)
modLotAreaAll=lm(Price~LotArea+I(LotArea^2)+I(sqrt(LotArea))+I(log(LotArea)), data=AmesTrain6a)
anova(modLotArea, modLotAreaSquared, modLotAreaSqrt, modLotAreaLog, modLotAreaAll)

plot(modLotArea$residuals~modLotArea$fitted.values)
abline(0,0)
plot(modLotAreaSquared$residuals~modLotAreaSquared$fitted.values)
abline(0,0)
```
This data shows that we should use LotArea^2 as our transformation because it has the best balance between the number of variables used and a significant p-value. 

```{r}
modYearBuilt=lm(Price~YearBuilt, data=AmesTrain6a)
modYearBuiltSquared=lm(Price~YearBuilt+I(YearBuilt^2), data=AmesTrain6a)
modYearBuiltSqrt=lm(Price~YearBuilt+I(sqrt(YearBuilt)), data=AmesTrain6a)
modYearBuiltLog=lm(Price~(log(YearBuilt)), data=AmesTrain6a)
modYearBuiltFull=lm(Price)~YearBuilt+I(YearBuilt^2)+I(sqrt(YearBuilt))+I(log(YearBuilt), data=AmesTrain6a)
anova(modYearBuilt, modYearBuiltSquared, modYearBuiltSqrt, modYearBuiltLog, modYearBuiltFull)

plot(modYearBuilt$residuals~modYearBuilt$fitted.values)
abline(0,0)
plot(modYearBuiltSquared$residuals~modYearBuiltSquared$fitted.values)
abline(0,0)
```
This shows that we should use YearBuilt^2 as our transformation because it has the best balance between the number of variables used and a significant p-value. 
```{r}
modWoodDeckSF=lm(Price~WoodDeckSF, data=AmesTrain6a)
modWoodDeckSFSquared=lm(Price~WoodDeckSF+I(WoodDeckSF^2), data=AmesTrain6a)
modWoodDeckSFSqrt=lm(Price~WoodDeckSF+I(sqrt(WoodDeckSF)), data=AmesTrain6a)
modWoodDeckSFLog=lm(Price~(log(WoodDeckSF+1)), data=AmesTrain6a)
modWoodDeckSFFull=lm(Price)~WoodDeckSF+I(WoodDeckSF^2)+I(sqrt(WoodDeckSF))+I(log(WoodDeckSF+1), data=AmesTrain6a)
anova(modWoodDeckSF, modWoodDeckSFSquared, modWoodDeckSFSqrt, modWoodDeckSFLog, modWoodDeckSFFull)

plot(modWoodDeckSF$residuals~modWoodDeckSF$fitted.values)
abline(0,0)
plot(modWoodDeckSFSquared$residuals~modWoodDeckSFSquared$fitted.values)
abline(0,0)
```
This shows that we should use WoodDeck^2 as our transformation because it has the best balance between the number of variables used and a significant p-value. 
```{r}
modGroundSF=lm(Price~GroundSF, data=AmesTrain6a)
modGroundSFSquared=lm(Price~GroundSF+I(GroundSF^2), data=AmesTrain6a)
modGroundSFSqrt=lm(Price~GroundSF+I(sqrt(GroundSF)), data=AmesTrain6a)
modGroundSFLog=lm(Price~(log(GroundSF+1)), data=AmesTrain6a)
modGroundSFFull=lm(Price)~GroundSFF+I(GroundSF^2)+I(sqrt(GroundSF))+I(log(GroundSF+1), data=AmesTrain6a)
anova(modGroundSF, modGroundSFSquared, modGroundSFSqrt, modGroundSFLog, modGroundSFFull)

plot(modGroundSF$residuals~modGroundSF$fitted.values)
abline(0,0)
plot(modGroundSFSquared$residuals~modGroundSFSquared$fitted.values)
abline(0,0)
```
We're keeping GroundSF as our transformation because it has the best balance between the number of variables used and a significant p-value.
```{r}
modFullBath = lm(Price~FullBath, data = AmesTrain6a)
modFullBathSquared= lm(Price ~ FullBath+I(FullBath^2), data = AmesTrain6a)
modFullBathSqrt= lm(Price~FullBath+I(sqrt(FullBath)), data=AmesTrain6a)
modFullBathLog= lm(Price~(log(FullBath+1)), data=AmesTrain6a)
modFullBathFull = lm(Price~FullBath+I(FullBath^2)+I(sqrt(FullBath))+I(log(FullBath+1)), data=AmesTrain6a)
anova(modFullBath, modFullBathSquared, modFullBathSqrt, modFullBathLog, modFullBathFull)

plot(modFullBath$residuals~modFullBath$fitted.values)
abline(0,0)
plot(modFullBathSquared$residuals~modFullBathSquared$fitted.values)
abline(0,0)
```
This shows that we should use FullBath^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.
```{r}
modTotalRooms = lm(Price ~ TotalRooms, data = AmesTrain6a)
modTotalRoomsSquared = lm(Price ~ TotalRooms+I(TotalRooms^2), data = AmesTrain6a)
modTotalRoomsSqrt = lm(Price~TotalRooms+I(sqrt(TotalRooms)), data=AmesTrain6a)
modTotalRoomsLog = lm(Price~(log(TotalRooms+1)), data=AmesTrain6a)
modTotalRoomsFull = lm(Price~TotalRooms+I(TotalRooms^2)+I(sqrt(TotalRooms))+I(log(TotalRooms+1)), data=AmesTrain6a)
anova(modTotalRooms, modTotalRoomsSquared, modTotalRoomsSqrt, modTotalRoomsLog, modTotalRoomsFull)

plot(modTotalRooms$residuals~modTotalRooms$fitted.values)
abline(0,0)
plot(modTotalRoomsSquared$residuals~modTotalRoomsSquared$fitted.values)
abline(0,0)
```
This shows that we should use TotalRooms + I(TotalRooms^2) + I(sqrt(TotalRooms)) + 
    I(log(TotalRooms + 1)) as our transformation because it is the only one with a signficant p-value. We may not include this variable in a final model because of how many variables it creates and relies on.
```{r}
modBasementSF=lm(Price~BasementSF, data=AmesTrain6a)
modBasementSFSquared=lm(Price~BasementSF+I(BasementSF^2), data=AmesTrain6a)
modBasementSFSqrt=lm(Price~BasementSF+I(sqrt(BasementSF)), data=AmesTrain6a)
modBasementSFLog=lm(Price~(log(BasementSF+1)), data=AmesTrain6a)
modBasementSFFull=lm(Price~BasementSF+I(BasementSF^2)+I(sqrt(BasementSF))+I(log(BasementSF+1)), data=AmesTrain6a)
anova(modBasementSF, modBasementSFSquared, modBasementSFSqrt, modBasementSFLog, modBasementSFFull)

plot(modBasementSF$residuals~modBasementSF$fitted.values)
abline(0,0)
plot(modBasementSFSquared$residuals~modBasementSFSquared$fitted.values)
abline(0,0)
```
This shows that we should use log(BasementSF) as our transformation because it has the best balance between the number of variables used and a significant p-value.
```{r}
modGarageCars=lm(Price~GarageCars, data=AmesTrain6a)
modGarageCarsSquared=lm(Price~GarageCars+I(GarageCars^2), data=AmesTrain6a)
modGarageCarsSqrt=lm(Price~GarageCars+I(sqrt(GarageCars)), data=AmesTrain6a)
modGarageCarsLog=lm(Price~(log(GarageCars+1)), data=AmesTrain6a)
modGarageCarsFull=lm(Price~GarageCars+I(GarageCars^2)+I(sqrt(GarageCars))+I(log(GarageCars+1)), data=AmesTrain6a)
anova(modGarageCars, modGarageCarsSquared, modGarageCarsSqrt, modGarageCarsLog, modGarageCarsFull)

plot(modGarageCars$residuals~modGarageCars$fitted.values)
abline(0,0)
plot(modGarageCarsSquared$residuals~modGarageCarsSquared$fitted.values)
abline(0,0)

```
This shows that we should use modGarageCars^2 as our transformation because it has the best balance between the number of variables used and a significant p-value.

#Transformations of the Response
```{r}
modTransformNumericLog=lm(log(Price)~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms, data=AmesTrain6a)
summary(modTransformNumericLog)
modTransformNumeric=lm(Price~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms, data=AmesTrain6a)
summary(modTransformNumeric)
modTransformCatLog=lm(log(Price)~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCatLog)
modTransformCat=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCat)
```
We decided not to log the response because the results of the logged Price were not significantly different. Although the adjusted r-squared was slightly better (.001) for log(Price) than Price with numeric variables, Price was significantly better than log(Price) for Categorical variables (.02). Additionally, we didn't want to overfit the data through too many transformations, so we chose to keep Price as the response variable. 

#Combinations of Variables

```{r}
modAllBathroom=lm(Price~FullBath+BasementFBath+0.5*BasementHBath+0.5*HalfBath, data-AmesTrain6a)
summary(modAllBathroom)
```
We chose not to combine any of the variables because they didn't significantly improve the model. For example, experimental combinations with the different bath variables didn't improve the adjusted r-squared value while also lowering AIC and Mallow Cp. 

#Final Selection for Fancy Model
```{r}
modTransformCat=lm(Price~factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(HeatingQC)+factor(KitchenQ)+factor(ExteriorC)+factor(CentralAir)+factor(GarageQ)+factor(Foundation)+factor(GarageC)+factor(BasementHt)+factor(GarageType)+factor(LotConfig)+factor(BasementC)+factor(Heating)+factor(Condition), data=AmesTrain6a)
summary(modTransformCat)
MSE=(summary(modTransformCat)$sigma)^2
step(none,scope=list(upper=modTransformCat),scale=MSE)
```

```{r}
modTransformNumeric=lm(Price~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms, data=AmesTrain6a)
summary(modTransformNumeric)
MSE=(summary(modTransformNumeric)$sigma)^2
step(none,scope=list(upper=modTransformNumeric),scale=MSE)
```
We chose to narrow down our pool of variables separately by categorical and numerical factors before we combined them. It was easier to analyze the numeric and categorical variables in models together. However, once we narrowed down the categorical and numerical variables seperately, we combined them in this model (modTransformFull) and re-ran stepwise, forward, and backward selection, which is below.



```{r}
modTransformFull=lm(Price~LotArea+I(LotArea^2)+YearBuilt+I(YearBuilt^2)+BasementSF+I(BasementSF^2)+GarageCars+I(GarageCars^2)+WoodDeckSF+I(WoodDeckSF^2)+GroundSF+I(GroundSF^2)+FullBath+TotalRooms+factor(HouseStyle)+factor(ExteriorQ)+factor(BasementFin)+factor(KitchenQ)+factor(BasementHt)+factor(Condition), data=AmesTrain6a)
summary(modTransformFull)
```
We made this model by combining the categorical and numerical variables suggested by our separate stepwise selections.

Stepwise selection:
```{r}
MSE=(summary(modTransformFull)$sigma)^2
step(none,scope=list(upper=modTransformFull),scale=MSE)
```

Forward selection
```{r}
MSE=(summary(modTransformFull)$sigma)^2
none=lm(Price~1,data=AmesTrain6a)
step(none,scope=list(upper=modTransformFull),scale=MSE, direction = "forward")
```

Backward selection
```{r}
MSE=(summary(modTransformFull)$sigma)^2
step(modTransformFull,scale=MSE)
```
Backward selection



```{r}
BackwardMod = lm(Price ~ LotArea + I(LotArea^2) + I(YearBuilt^2) + BasementSF + 
    GarageCars + I(GarageCars^2) + WoodDeckSF + GroundSF + I(GroundSF^2) + 
    FullBath + TotalRooms + factor(HouseStyle) + factor(ExteriorQ) + 
    factor(BasementFin) + factor(KitchenQ) + factor(BasementHt) + 
    factor(Condition), data=AmesTrain6a)
summary(BackwardMod)
```

```{r}
StepwiseMod = lm(Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars + WoodDeckSF, data=AmesTrain6a)
summary(StepwiseMod)
```

```{r}
ForwardMod= lm(Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars, data=AmesTrain6a)
summary(ForwardMod)
```

The forward selection model has the best parsimony and suggests a 16-variable mod with an r-squared value of .887, an AIC of 40.22, and a Mallow CP of ~40. (This is one fewer variable but a slightly worse (.002) adjusted r-squared than the models suggested by stepwise and backward selection.) We chose the backward selection model because it uses the fewest variables, which is best for parsimony, especially given how many variables we could use. 

### Part 8: Cross-Validation
#Redo the cross-validation analysis with your test data for your new fancy model.

```{r}
mean(ForwardMod$residuals)
sd(ForwardMod$residuals)

mean(modTransformFull$residuals)
sd(modTransformFull$residuals)

plot(ForwardMod$residuals)
abline(0,0)
```
```{r}
ShrunkenMod=lm(Price ~ factor(ExteriorQ) + I(GroundSF^2) + factor(BasementHt) + 
    factor(HouseStyle) + factor(Condition) + I(YearBuilt^2) + 
    BasementSF + LotArea + I(LotArea^2) + factor(KitchenQ) + 
    I(GarageCars^2) + factor(BasementFin) + FullBath + GroundSF + 
    TotalRooms + GarageCars, data=AmesTrain6)

RefitAmes=predict.lm(ShrunkenMod, newdata=AmesTest6)

cor(AmesTest6$Price,RefitAmes)
crosscorr=cor(AmesTest6$Price,RefitAmes)
cor(log(AmesTest6$Price),RefitAmes)
crosscorr=cor(AmesTest6$Price,RefitAmes)
crosscorr^2
.8872-crosscorr^2
```
#Discuss mean of residuals, std. dev of residuals, cross-validation correlation, and shrinkage
The chunk beginning at line 312 shows us that having a model with fewer variables (ForwardMod) does not dramatically impact the residual values of either the mean or the standard deviation. The mean residuals of the ForwardMod is very close to zero, indicating that there is normality in the data. The plot of the residuals backs up this assumption, since it doesn't show any fanning pattern or skew. The standard deviation is also small relative to the size of the data. These are all good signs and encourage us to use ForwardMod. ForwardMod is a also a more efficent model than the full mod because it has fewer variables for essentially the same r-squared value (.002 difference). The cross-validation of the model also shows shrinkage of the ForwardMod as a healthy sign, since it shows only a .05 difference between the model we made and the holdout data. This means we did not overfit the model.  

###Part 9: Final Model
#Final Changes
We chose not to make any more adjustments to our model, because we think it does a good job balancing the number of variables it uses and predictive ability. This conclusion was supported by normality in the residuals and negative shrinkage. The adjusted r-squared is high and we have relatively good parsimony. 

```{r}
newpredictiondata= data.frame(ExteriorQ="Gd", BasementHt="Ex", Condition=5, YearBuilt=1995, BasementSF=1150, KitchenQ="TA", GarageCars=2, BasementFin="Unf", TotalRooms=9, GroundSF=2314, FullBath=2, HouseStyle="2Story", WoodDeckSF=274, LotArea=11060)
predict.lm(ForwardMod, newpredictiondata, interval="prediction", level=.95)
```

With a 2 story house from Ames, Iowa, with a good exterior quality, excellent basement height, average overall condition, built in 1995, basement square footage of 1150 ft, average kitchen quality, space for 2 cars in the garage, unfinished basement, 9 total rooms, 2314 ft in living area square feet, 2 full baths, 274 sq ft of wood deck, and 11060 sq ft lot area, we expect the price to be $234,182. We are 95% confident that the price will fall between $178,715 and $289,649. 

